8471:
8993:
8192:
505:
748:
13672:
38:
8100:
13164:
4285:. In these characteristics, if the derivative is not a constant, it is a linear polynomial in characteristic 3, and is the square of a linear polynomial in characteristic 2. Therefore, for either characteristic 2 or 3, the derivative has only one root. This allows computing the multiple root, and the third root can be deduced from the sum of the roots, which is provided by
12378:
799:(1501–1576) to reveal his secret for solving cubic equations. In 1539, Tartaglia did so only on the condition that Cardano would never reveal it and that if he did write a book about cubics, he would give Tartaglia time to publish. Some years later, Cardano learned about del Ferro's prior work and published del Ferro's method in his book
8723:
7728:
12070:
1650:
13667:{\displaystyle {\begin{aligned}P&=s_{1}s_{2}=x_{0}^{2}+x_{1}^{2}+x_{2}^{2}-(x_{0}x_{1}+x_{1}x_{2}+x_{2}x_{0}),\\S&=s_{1}^{3}+s_{2}^{3}=2(x_{0}^{3}+x_{1}^{3}+x_{2}^{3})-3(x_{0}^{2}x_{1}+x_{1}^{2}x_{2}+x_{2}^{2}x_{0}+x_{0}x_{1}^{2}+x_{1}x_{2}^{2}+x_{2}x_{0}^{2})+12x_{0}x_{1}x_{2}.\end{aligned}}}
5183:
of the root function (that is the root that has the largest real part). With this convention
Cardano's formula for the three roots remains valid, but is not purely algebraic, as the definition of a principal part is not purely algebraic, since it involves inequalities for comparing real parts. Also,
606:(1048–1131), made significant progress in the theory of cubic equations. In an early paper, he discovered that a cubic equation can have more than one solution and stated that it cannot be solved using compass and straightedge constructions. He also found a geometric solution. In his later work, the
808:
Cardano's promise to
Tartaglia said that he would not publish Tartaglia's work, and Cardano felt he was publishing del Ferro's, so as to get around the promise. Nevertheless, this led to a challenge to Cardano from Tartaglia, which Cardano denied. The challenge was eventually accepted by Cardano's
12102:
763:
and announced that he could solve them. He was soon challenged by Fior, which led to a famous contest between the two. Each contestant had to put up a certain amount of money and to propose a number of problems for his rival to solve. Whoever solved more problems within 30 days would get all the
14269:. The geometric construction was perfectly suitable for Omar Khayyam, as it occurs for solving a problem of geometric construction. At the end of his article he says only that, for this geometrical problem, if approximations are sufficient, then a simpler solution may be obtained by consulting
10765:
10160:
9126:
The points in the complex plane representing the three roots serve as the vertices of an isosceles triangle. (The triangle is isosceles because one root is on the horizontal (real) axis and the other two roots, being complex conjugates, appear symmetrically above and below the real axis.)
4280:
The reduction to a depressed cubic works for characteristic 2, but not for characteristic 3. However, in both cases, it is simpler to establish and state the results for the general cubic. The main tool for that is the fact that a multiple root is a common root of the polynomial and its
8546:
227:
5319:
6582:
14024:
can be expressed in terms of the roots of a cubic. Further, the ratios of the long diagonal to the side, the side to the short diagonal, and the negative of the short diagonal to the long diagonal all satisfy a particular cubic equation. In addition, the ratio of the
13127:
In other words, in this case, Cardano's method and
Lagrange's method compute exactly the same things, up to a factor of three in the auxiliary variables, the main difference being that Lagrange's method explains why these auxiliary variables appear in the problem.
11657:, because it requires solving a resolvent polynomial of degree at least six. Apart from the fact that nobody had previously succeeded, this was the first indication of the non-existence of an algebraic formula for degrees 5 and higher; as was later proved by the
5857:
13834:
14708:
8095:{\displaystyle {\begin{aligned}t_{0}&=-2{\frac {|q|}{q}}{\sqrt {-{\frac {p}{3}}}}\cosh \left\qquad {\text{if }}~4p^{3}+27q^{2}>0~{\text{ and }}~p<0,\\t_{0}&=-2{\sqrt {\frac {p}{3}}}\sinh \left\qquad {\text{if }}~p>0.\end{aligned}}}
11848:
1457:
437:(20th to 16th centuries BC) cuneiform tablets have been found with tables for calculating cubes and cube roots. The Babylonians could have used the tables to solve cubic equations, but no evidence exists to confirm that they did. The problem of
6111:
1192:
3569:
10568:
10475:
7447:
9885:
10608:
9931:
7705:
7074:
420:
other than 2 and 3. The solutions of the cubic equation do not necessarily belong to the same field as the coefficients. For example, some cubic equations with rational coefficients have roots that are irrational (and even non-real)
11388:
6399:
10357:
4865:
9122:
With one real and two complex roots, the three roots can be represented as points in the complex plane, as can the two roots of the cubic's derivative. There is an interesting geometrical relationship among all these roots.
4701:
4624:
2991:
9393:
2603:
5024:
11816:
5202:
4943:
9199:, the triangle is equilateral, the Steiner inellipse is simply the triangle's incircle, its foci coincide with each other at the incenter, which lies on the real axis, and hence the derivative has duplicate real roots.
8795:
12494:
are also symmetric in the roots of the cubic equation to be solved. Thus these symmetric functions can be expressed in terms of the (known) coefficients of the original cubic, and this allows eventually expressing the
8373:
7526:
2136:
2462:
and the discriminant of the corresponding depressed cubic. Using the formula relating the general cubic and the associated depressed cubic, this implies that the discriminant of the general cubic can be written as
608:
2433:
6442:
3875:
12373:{\displaystyle {\begin{aligned}x_{0}&={\tfrac {1}{3}}(s_{0}+s_{1}+s_{2}),\\x_{1}&={\tfrac {1}{3}}(s_{0}+\xi ^{2}s_{1}+\xi s_{2}),\\x_{2}&={\tfrac {1}{3}}(s_{0}+\xi s_{1}+\xi ^{2}s_{2}).\end{aligned}}}
5392:
denote any square root and any cube root. The other roots of the equation are obtained either by changing of cube root or, equivalently, by multiplying the cube root by a primitive cube root of unity, that is
7264:
4138:
62:
14267:
and he found a positive root of this cubic by considering the intersection of a rectangular hyperbola and a circle. An approximate numerical solution was then found by interpolation in trigonometric tables
4786:
11306:
5441:
4292:
A difference with other characteristics is that, in characteristic 2, the formula for a double root involves a square root, and, in characteristic 3, the formula for a triple root involves a cube root.
4259:
4022:
5724:
4431:
13693:
6761:
13698:
13169:
12107:
11853:
11206:
9936:
9801:
8718:{\displaystyle t_{k}=2{\sqrt {-{\frac {p}{3}}}}\cos \left({\frac {1}{3}}\arccos \left({\frac {3q}{2p}}{\sqrt {\frac {-3}{p}}}\right)-k{\frac {2\pi }{3}}\right)\quad {\text{for}}\quad k=0,1,2\,.}
7733:
5729:
3770:
1462:
67:
12885:
10280:
11569:
7312:
2244:
939:
Finding the roots of a reducible cubic equation is easier than solving the general case. In fact, if the equation is reducible, one of the factors must have degree one, and thus have the form
11037:
1054:
10222:
4496:
3445:
655:), which dealt with eight types of cubic equations with positive solutions and five types of cubic equations which may not have positive solutions. He used what would later be known as the
9751:
12509:
as roots has a degree higher than that of the initial polynomial, and is therefore unhelpful for solving. This is the reason for which
Lagrange's method fails in degrees five and higher.
1822:
8882:
6807:
6662:
6298:
8516:
Viète's trigonometric expression of the roots in the three-real-roots case lends itself to a geometric interpretation in terms of a circle. When the cubic is written in depressed form
6021:
13023:
12612:
3374:
1063:
922:
10878:
10823:
5683:
5081:
3452:
1337:
310:
10978:
10932:
10479:
10386:
7354:
7349:
7132:
1998:
1381:
9796:
3647:
2850:
2307:
707:
he gave the result as 1,22,7,42,33,4,40 (equivalent to 1 + 22/60 + 7/60 + 42/60 + 33/60 + 4/60 + 40/60), which has a
12731:
9472:
9448:
7531:
6897:
6702:
6144:
5579:
5352:
5139:
3322:
3252:
1761:
1708:
6182:
5390:
5177:
624:
In the 12th century, the Indian mathematician
Bhaskara II attempted the solution of cubic equations without general success. However, he gave one example of a cubic equation:
12937:
11649:
introduced a new method to solve equations of low degree in a uniform way, with the hope that he could generalize it for higher degrees. This method works well for cubic and
3919:
12556:
6242:
3144:
3019:
2743:
2708:
1452:
11313:
10884:
has three cube roots, using
Cardano's formula without care would provide nine roots, while a cubic equation cannot have more than three roots. This was clarified first by
6892:
6319:
4365:
489:, though historians such as Reviel Netz dispute whether the Greeks were thinking about cubic equations or just problems that can lead to cubic equations. Some others like
10285:
5992:
5944:
5913:
5886:
14275:
If the seeker is satisfied with an estimate, it is up to him to look into the table of chords of
Almagest, or the table of sines and versed sines of Mothmed Observatory.
3693:
3200:
2180:
1263:
13125:
12976:
12676:
12644:
13086:
9791:
14066:
2855:
2675:
1860:
10603:
9923:
9271:
2466:
12785:
12758:
9660:
4550:
4523:
3120:
are complex conjugates, and their product is real and positive. Thus the discriminant is the product of a single negative number and several positive ones. That is
2460:
14976:
New and Easy Method of
Solution of the Cubic and Biquadratic Equations: Embracing Several New Formulas, Greatly Simplifying this Department of Mathematical Science
5497:
966:
5523:
2761:
is also a root. So the non-real roots, if any, occur as pairs of complex conjugate roots. As a cubic polynomial has three roots (not necessarily distinct) by the
13050:
12812:
12065:{\displaystyle {\begin{aligned}s_{0}&=x_{0}+x_{1}+x_{2},\\s_{1}&=x_{0}+\xi x_{1}+\xi ^{2}x_{2},\\s_{2}&=x_{0}+\xi ^{2}x_{1}+\xi x_{2},\end{aligned}}}
11664:
In the case of cubic equations, Lagrange's method gives the same solution as
Cardano's. Lagrange's method can be applied directly to the general cubic equation
1645:{\displaystyle {\begin{aligned}t={}&x+{\frac {b}{3a}}\\p={}&{\frac {3ac-b^{2}}{3a^{2}}}\\q={}&{\frac {2b^{3}-9abc+27a^{2}d}{27a^{3}}}.\end{aligned}}}
10381:
9628:
8295:
7452:
2003:
14697:, p. 9) states, "Omar Al Hay of Chorassan, about 1079 AD did most to elevate to a method the solution of the algebraic equations by intersecting conics."
7134:). Otherwise, it is still correct but involves complex cosines and arccosines when there is only one real root, and it is nonsensical (division by zero) when
15461:
2312:
12502:
as roots of a polynomial with known coefficients. This works well for every degree, but, in degrees higher than four, the resulting polynomial that has the
6316:
occurs in following formulas; this fraction must be interpreted as equal to zero (see the end of this section). With these conventions, one of the roots is
3775:
441:
involves the simplest and oldest studied cubic equation, and one for which the ancient
Egyptians did not believe a solution existed. In the 5th century BC,
9139:
of the triangle—the unique ellipse that is tangent to the triangle at the midpoints of its sides. If the angle at the vertex on the real axis is less than
816:
Cardano noticed that Tartaglia's method sometimes required him to extract the square root of a negative number. He even included a calculation with these
15528:
5184:
the use of principal cube root may give a wrong result if the coefficients are non-real complex numbers. Moreover, if the coefficients belong to another
4791:
6615:
As for the special case of a depressed cubic, this formula applies but is useless when the roots can be expressed without cube roots. In particular, if
5091:
involving only real numbers. Therefore, the equation cannot be solved in this case with the knowledge of Cardano's time. This case has thus been called
493:, who translated all of Archimedes's works, disagree, putting forward evidence that Archimedes really solved cubic equations using intersections of two
7191:
4629:
4027:
744:
were not known to him at that time. Del Ferro kept his achievement secret until just before his death, when he told his student Antonio Fior about it.
15133:
4555:
7144:
This formula can be straightforwardly transformed into a formula for the roots of a general cubic equation, using the back-substitution described in
6821:
When a cubic equation with real coefficients has three real roots, the formulas expressing these roots in terms of radicals involve complex numbers.
4948:
1265:
are said to be depressed. They are much simpler than general cubics, but are fundamental, because the study of any cubic may be reduced by a simple
11738:
10760:{\displaystyle t={\sqrt{-{q \over 2}+{\sqrt {{q^{2} \over 4}+{p^{3} \over 27}}}}}+{\sqrt{-{q \over 2}-{\sqrt {{q^{2} \over 4}+{p^{3} \over 27}}}}}}
10155:{\displaystyle {\begin{aligned}0&=(x-u^{3})(x-v^{3})\\&=x^{2}-(u^{3}+v^{3})x+u^{3}v^{3}\\&=x^{2}-(u^{3}+v^{3})x+(uv)^{3}\end{aligned}}}
4870:
8730:
1898:, or, if it is divisible by the square of a non-constant polynomial. In other words, the discriminant is nonzero if and only if the polynomial is
11223:
4143:
3924:
760:
445:
reduced this problem to that of finding two mean proportionals between one line and another of twice its length, but could not solve this with a
12099:
are known, the roots may be recovered from them with the inverse Fourier transform consisting of inverting this linear transformation; that is,
9159:
then the major axis of the ellipse lies on the real axis, as do its foci and hence the roots of the derivative. If that angle is greater than
8109:
and the inequalities on the right are not satisfied (the case of three real roots), the formulas remain valid but involve complex quantities.
3154:
10987:
The other roots of the equation can be obtained by changing of cube root, or, equivalently, by multiplying the cube root by each of the two
8275:-axis. The solution is given by the length of the horizontal line segment from the origin to the intersection of the vertical line and the
14738:
11140:
3698:
222:{\displaystyle {\begin{aligned}f(x)&={\frac {1}{4}}\left(x^{3}+3x^{2}-6x-8\right)\\&={\frac {1}{4}}(x-2)(x+1)(x+4)\end{aligned}}}
14204:
The steady state speed of a vehicle moving on a slope with air friction for a given input power is solved by a depressed cubic equation.
6831:), one cannot express the roots in terms of real radicals. Nevertheless, purely real expressions of the solutions may be obtained using
2611:, the two discriminants have the same sign. In summary, the same information can be deduced from either one of these two discriminants.
805:
in 1545, meaning Cardano gave Tartaglia six years to publish his results (with credit given to Tartaglia for an independent solution).
2185:
9082:-coordinate) of the horizontal intercept of the curve (point R on the figure). Further, if the complex conjugate roots are written as
5314:{\displaystyle C-{\frac {p}{3C}}\quad {\text{with}}\quad C={\sqrt{-{\frac {q}{2}}+{\sqrt {{\frac {q^{2}}{4}}+{\frac {p^{3}}{27}}}}}}.}
14207:
10165:
451:
4730:
4440:
3381:
14181:
11618:, and therefore does not change the roots. This method only fails when both roots of the quadratic equation are zero, that is when
9665:
1766:
14835:
14773:
14682:
14239:
5396:
5108:, Cardano's formula can still be used, but some care is needed in the use of cube roots. A first method is to define the symbols
6577:{\displaystyle x_{k}=-{\frac {1}{3a}}\left(b+\xi ^{k}C+{\frac {\Delta _{0}}{\xi ^{k}C}}\right),\qquad k\in \{0,1,2\}{\text{,}}}
620:“We have tried to express these roots by algebra but have failed. It may be, however, that men who come after us will succeed.”
15922:
15760:
15658:
15540:
15439:
15409:
14718:
14654:
14537:
14510:
14477:
14431:
14402:
13988:
8431:
446:
6204:"; the choice is almost arbitrary, and changing it amounts to choosing a different square root. However, if a choice yields
4382:
8971:, without changing the angle relationships. This shift moves the point of inflection and the centre of the circle onto the
5525:
Similarly, the formula is also useless in the cases where no cube root is needed, that is when the cubic polynomial is not
3334:
2607:
It follows that one of these two discriminants is zero if and only if the other is also zero, and, if the coefficients are
857:
6707:
5621:
1275:
248:
9220:
8470:
1342:
12819:
10282:, and assuming it is positive, real solutions to this equation are (after folding division by 4 under the square root):
10227:
5191:
The second way for making Cardano's formula always correct, is to remark that the product of the two cube roots must be
813:(1522–1565). Ferrari did better than Tartaglia in the competition, and Tartaglia lost both his prestige and his income.
612:, he wrote a complete classification of cubic equations with general geometric solutions found by means of intersecting
16102:
14626:
14598:
14105:
11527:
7269:
5715:
other than 2 or 3. If the coefficients are real numbers, the formula covers all complex solutions, not just real ones.
673:
of curves in order to solve cubic equations which may not have positive solutions. He understood the importance of the
14555:, p. 8) states that "the Egyptians considered the solution impossible, but the Greeks came nearer to a solution."
10994:
1014:
15477:
15361:
15090:
These are Formulas (80) and (83) of Weisstein, Eric W. 'Cubic Formula'. From MathWorld—A Wolfram Web Resource.
15075:
14984:
14868:
14748:
14454:
9179:, the major axis is vertical and its foci, the roots of the derivative, are complex conjugates. And if that angle is
8441:
A cubic equation can be solved by compass-and-straightedge construction (without trisector) if and only if it has a
15353:
14974:
1406:
11050:(Vieta is his Latin name) in a text published posthumously in 1615, which provides directly the second formula of
8992:
6849:
4322:
15345:
14270:
13687:
5852:{\displaystyle {\begin{aligned}\Delta _{0}&=b^{2}-3ac,\\\Delta _{1}&=2b^{3}-9abc+27a^{2}d.\end{aligned}}}
4277:
other than 2 or 3, but must be modified for characteristic 2 or 3, because of the involved divisions by 2 and 3.
8850:
6766:
6618:
6254:
1226:
412:
The coefficients do not need to be real numbers. Much of what is covered below is valid for coefficients in any
14890:
2762:
14177:
13829:{\displaystyle {\begin{aligned}P&=e_{1}^{2}-3e_{2},\\S&=2e_{1}^{3}-9e_{1}e_{2}+27e_{3},\end{aligned}}}
15866:
15277:
15241:
11661:. Nevertheless, modern methods for solving solvable quintic equations are mainly based on Lagrange's method.
8963:. Graphically this corresponds to simply shifting the graph horizontally when changing between the variables
6192:
cube root, respectively (every nonzero complex number has two square roots and three cubic roots). The sign "
929:
640:
15873:
15239:
Henriquez, Garcia (June–July 1935), "The graphical interpretation of the complex roots of cubic equations",
644:
16097:
16081:
15164:
14130:
14113:
12981:
12561:
9583:
6763:
A straightforward computation allows verifying that the existence of this factorization is equivalent with
801:
56:
10828:
10773:
9575:
8191:
5031:
942:
16153:
16117:
15861:
14685:, states, "Khayyam himself seems to have been the first to conceive a general theory of cubic equations."
10937:
10895:
7317:
7085:
1939:
14768:
14037:
is one of the solutions of a cubic equation. The values of trigonometric functions of angles related to
5689:. The variant that is presented here is valid not only for real coefficients, but also for coefficients
3591:
2794:
2251:
850:, one can obtain an equivalent equation with integer coefficients, by multiplying all coefficients by a
15962:
12681:
11702:
9453:
9429:
6667:
6116:
5532:
5324:
5111:
3272:
3205:
1713:
1660:
14198:
8897:(shown in the accompanying graph), the depressed case as indicated previously is obtained by defining
6149:
5357:
5144:
834:(1540–1603) independently derived the trigonometric solution for the cubic with three real roots, and
15915:
15856:
14109:
14098:
12890:
8157:, which is the proper Chebyshev cube root. The value involving hyperbolic sines is similarly denoted
4309:
828:
studied this issue in detail and is therefore often considered as the discoverer of complex numbers.
616:. Khayyam made an attempt to come up with an algebraic formula for extracting cubic roots. He wrote:
9243:). As these automorphisms must permute the roots of the polynomials, this group is either the group
9114:
are the square roots of the tangent of the angle between this tangent line and the horizontal axis.
3882:
3573:
This result can be proved by expanding the latter product or retrieved by solving the rather simple
16107:
15818:
15778:
15716:
15676:
15607:
15559:
15275:
Barr, C. F. (1918), "Discussions: Relating to the Graph of a Cubic Equation Having Complex Roots",
14840:
14830:
14778:
14244:
12515:
11658:
9216:
8271:-axis, and a vertical line through the point where the circle and the parabola intersect above the
6214:
5712:
5596:
4274:
3588:, these results can be extended to the general cubic. This gives: If the discriminant of the cubic
3123:
2998:
2995:
If the three roots are real and distinct, the discriminant is a product of positive reals, that is
2719:
2684:
1057:
657:
417:
386:
14381:
Høyrup, Jens (1992), "The Babylonian Cellar Text BM 85200 + VAT 6599 Retranslation and Analysis",
10980:
This means that only one cube root needs to be computed, and leads to the second formula given in
16046:
14590:
14234:
11710:
10988:
6412:
6403:
The other two roots can be obtained by changing the choice of the cube root in the definition of
5970:
5922:
5891:
5864:
4725:
3067:
397:
15769:
Rechtschaffen, Edgar (July 2008), "Real roots of cubics: Explicit formula for quasi-solutions",
14005:
of any triangle can be found by using the cubic function whose roots are the coordinates in the
6106:{\displaystyle C={\sqrt{\frac {\Delta _{1}\pm {\sqrt {\Delta _{1}^{2}-4\Delta _{0}^{3}}}}{2}}},}
3656:
3163:
2143:
433:
Cubic equations were known to the ancient Babylonians, Greeks, Chinese, Indians, and Egyptians.
16076:
16071:
16066:
15944:
14930:
13683:
13091:
12942:
12649:
12617:
9208:
5526:
1899:
485:
are believed to have come close to solving the problem of doubling the cube using intersecting
401:
15894:
about the history of cubic equations and Cardano's solution, as well as Ferrari's solution to
15416:
14860:
14854:
14502:
14421:
13055:
9758:
1187:{\displaystyle {\frac {a}{q}}x^{2}+{\frac {bq+ap}{q^{2}}}x+{\frac {cq^{2}+bpq+ap^{2}}{q^{3}}}}
16056:
16036:
15376:
14678:
14527:
14467:
14040:
11646:
4304:
is credited with publishing the first formula for solving cubic equations, attributing it to
3564:{\displaystyle t^{3}+pt+q=\left(t-{\frac {3q}{p}}\right)\left(t+{\frac {3q}{2p}}\right)^{2}.}
2660:
1827:
366:
10573:
10563:{\displaystyle v={\sqrt{-{\frac {q}{2}}-{\sqrt {{\frac {q^{2}}{4}}+{\frac {p^{3}}{27}}}}}}.}
10470:{\displaystyle u={\sqrt{-{\frac {q}{2}}+{\sqrt {{\frac {q^{2}}{4}}+{\frac {p^{3}}{27}}}}}}.}
9898:
7442:{\displaystyle 4\cos ^{3}\theta -3\cos \theta -{\frac {3q}{2p}}\,{\sqrt {\frac {-3}{p}}}=0.}
718:(1465–1526) found a method for solving a class of cubic equations, namely those of the form
449:, a task which is now known to be impossible. Methods for solving cubic equations appear in
16163:
16122:
16041:
15908:
15380:
15105:
15029:
14938:
14734:
14158:
14094:
12763:
12736:
12733:
Thus the resolution of the equation may be finished exactly as with Cardano's method, with
12437:
9880:{\displaystyle {\begin{aligned}u^{3}+v^{3}&=-q\\uv&=-{\frac {p}{3}}.\end{aligned}}}
9633:
8501:
5088:
4528:
4501:
2438:
1208:
928:
if the polynomial on the left-hand side is the product of polynomials of lower degrees. By
925:
474:
442:
15182:
7700:{\displaystyle t_{k}=2\,{\sqrt {-{\frac {p}{3}}}}\,\cos \left\qquad {\text{for }}k=0,1,2.}
7069:{\displaystyle t_{k}=2\,{\sqrt {-{\frac {p}{3}}}}\,\cos \left\qquad {\text{for }}k=0,1,2.}
5473:
756:
8:
15935:
14826:
14791:
Berggren, J. L. (1990), "Innovation and Tradition in Sharaf al-Dīn al-Ṭūsī's Muʿādalāt",
14764:
14584:
14580:
14230:
14117:
14034:
13994:
12383:
11047:
9128:
7722:
7079:
6827:
5708:
5592:
5502:
5185:
5093:
4286:
4270:
3578:
3574:
2615:
831:
704:
688:(1170–1250), was able to closely approximate the positive solution to the cubic equation
508:
456:
413:
351:
14127:
and straight line can be computed using direct cubic equation representing Bézier curve.
14071:
Given the cosine (or other trigonometric function) of an arbitrary angle, the cosine of
13032:
12794:
16158:
16112:
15979:
15974:
15900:
15838:
15798:
15736:
15696:
15635:
15627:
15587:
15579:
15509:
15328:
15294:
15258:
15221:
15213:
15156:
15046:
14955:
14907:
14808:
14449:
Van der Waerden, Geometry and Algebra of Ancient Civilizations, chapter 4, Zurich 1983
10366:
9926:
9613:
9571:
9228:
5087:
allows proving that, if there is no rational root, the roots cannot be expressed by an
4305:
2711:
1929:
1871:
1655:
1266:
981:
715:
677:
of the cubic equation to find algebraic solutions to certain types of cubic equations.
666:
662:
560:
405:
378:
320:
45:
15809:
Zucker, I. J. (July 2008), "The cubic equation – a new look at the irreducible case",
9107:-intercept R of the cubic (that is the signed length OM, negative on the figure). The
15957:
15895:
15842:
15830:
15802:
15790:
15756:
15740:
15728:
15700:
15688:
15654:
15639:
15619:
15591:
15571:
15536:
15473:
15435:
15405:
15357:
15225:
15205:
15071:
15065:
15050:
15027:
Zucker, I.J. (July 2008). "The cubic equation — a new look at the irreducible case".
14980:
14959:
14911:
14864:
14744:
14714:
14650:
14622:
14594:
14533:
14506:
14495:
14473:
14450:
14427:
14398:
14002:
13980:
11383:{\displaystyle W=-{\frac {q}{2}}\pm {\sqrt {{\frac {p^{3}}{27}}+{\frac {q^{2}}{4}}}}}
9793:
This removes the third term in previous equality, leading to the system of equations
9136:
8996:
The slope of line RA is twice that of RH. Denoting the complex roots of the cubic as
8847:
finds the other cube roots; and multiplying the cosines of these resulting angles by
8454:
8450:
8140:
6394:{\displaystyle x=-{\frac {1}{3a}}\left(b+C+{\frac {\Delta _{0}}{C}}\right){\text{.}}}
4717:
4282:
2758:
2746:
1212:
670:
467:
438:
24:
15884:
500 years of NOT teaching THE CUBIC FORMULA. What is it they think you can't handle?
10352:{\displaystyle -{\frac {q}{2}}\pm {\sqrt {{\frac {q^{2}}{4}}+{\frac {p^{3}}{27}}}}.}
5718:
The formula being rather complicated, it is worth splitting it in smaller formulas.
4724:) that the two other roots are obtained by multiplying one of the cube roots by the
835:
792:, which proved to be too difficult for him to solve, and Tartaglia won the contest.
778:, for which he had worked out a general method. Fior received questions in the form
16010:
16003:
15998:
15877:
15822:
15782:
15748:
15720:
15680:
15611:
15563:
15501:
15320:
15286:
15250:
15197:
15148:
15038:
14947:
14899:
14800:
14390:
14162:
14144:
14140:
14134:
14079:
14072:
14010:
13998:
13976:
11654:
11650:
9579:
9132:
9131:
says that the points representing the roots of the derivative of the cubic are the
8509:
8446:
8435:
4860:{\displaystyle \varepsilon _{2}=\varepsilon _{1}^{2}={\frac {-1-i{\sqrt {3}}}{2}}.}
4301:
1894:
is a function of its coefficients that is zero if and only if the polynomial has a
977:
810:
796:
382:
16132:
15091:
14353:, and differs from the substitution that is used here only by a change of sign of
6015:
times the resultant of the first and second derivatives of the cubic polynomial.)
16020:
16015:
15967:
15952:
15646:
15446:
15429:
15399:
15120:
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables
14394:
14166:
14083:
10885:
10605:
the sum of the cube roots of these solutions is a root of the equation. That is
9506:
9240:
9071:
8442:
8132:
More precisely, the values involving cosines and hyperbolic cosines define, when
5180:
4696:{\displaystyle -{\frac {q}{2}}-{\sqrt {{\frac {q^{2}}{4}}+{\frac {p^{3}}{27}}}}.}
2645:. The proof then results in the verification of the equality of two polynomials.
851:
847:
825:
741:
14124:
8246:
4619:{\displaystyle -{\frac {q}{2}}+{\sqrt {{\frac {q^{2}}{4}}+{\frac {p^{3}}{27}}}}}
2986:{\displaystyle \Delta =a^{4}(r_{1}-r_{2})^{2}(r_{1}-r_{3})^{2}(r_{2}-r_{3})^{2}}
15986:
14277:
This is followed by a short description of this alternate method (seven lines).
14191:
14173:
14147:
are the solution of a cubic equation (the second derivative set equal to zero).
11732:
10881:
9388:{\displaystyle {\sqrt {\Delta }}=a^{2}(r_{1}-r_{2})(r_{1}-r_{3})(r_{2}-r_{3}),}
9108:
9067:
4721:
2598:{\displaystyle {\frac {4(b^{2}-3ac)^{3}-(2b^{3}-9abc+27a^{2}d)^{2}}{27a^{2}}}.}
817:
708:
486:
422:
354:). All of the roots of the cubic equation can be found by the following means:
324:
31:
15883:
15826:
15786:
15724:
15684:
15042:
14951:
5019:{\displaystyle \varepsilon _{2}{\sqrt{u_{1}}}+\varepsilon _{1}{\sqrt{u_{2}}}.}
19:
This article is about cubic equations in one variable. For cubic equations in
16147:
15834:
15794:
15732:
15692:
15623:
15575:
15492:
Guilbeau, Lucye (1930), "The History of the Solution of the Cubic Equation",
15311:
Irwin, Frank; Wright, H. N. (1917), "Some Properties of Polynomial Curves.",
15209:
15103:
Holmes, G. C., "The use of hyperbolic cosines in solving cubic polynomials",
14642:
14006:
11811:{\displaystyle \textstyle \xi ={\frac {-1\pm i{\sqrt {3}}}{2}}=e^{2i\pi /3},}
8217:-axis at the center of the circle is happenstance of the example illustrated.
6822:
5084:
4938:{\displaystyle \varepsilon _{1}{\sqrt{u_{1}}}+\varepsilon _{2}{\sqrt{u_{2}}}}
1895:
1874:
of a cubic can be determined without computing them explicitly, by using the
1056:
and the other roots are the roots of the other factor, which can be found by
933:
613:
556:
552:
494:
14386:
8815:
of that angle corresponds to taking a cube root of a complex number; adding
8790:{\displaystyle \arccos \left({\frac {3q}{2p}}{\sqrt {\frac {-3}{p}}}\right)}
3157:. If furthermore its coefficients are real, then all of its roots are real.
504:
16127:
15747:
Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. (2007),
14180:(which relate pressure, volume, and temperature of a substances), e.g. the
14030:
9100:
8264:
6832:
4434:
1887:
1875:
674:
603:
548:
392:
328:
14383:
Amphora: Festschrift for Hans Wussing on the Occasion of his 65th Birthday
8430:
A cubic equation with real coefficients can be solved geometrically using
8368:{\displaystyle {\frac {x^{4}}{m^{2}}}=x\left({\frac {n}{m^{2}}}-x\right).}
15465:
14885:
14137:
are found by solving a cubic equation (the derivative set equal to zero).
7521:{\displaystyle \cos(3\theta )={\frac {3q}{2p}}{\sqrt {\frac {-3}{p}}}\,,}
6825:
allows proving that when the three roots are real, and none is rational (
5685:
can be deduced from every variant of Cardano's formula by reduction to a
4376:
2654:
2608:
2131:{\displaystyle a^{4}(r_{1}-r_{2})^{2}(r_{1}-r_{3})^{2}(r_{2}-r_{3})^{2}.}
1933:
747:
732:. In fact, all cubic equations can be reduced to this form if one allows
490:
370:
347:
42:
15707:
Mitchell, D. W. (November 2009), "Powers of φ as roots of cubics",
15667:
Mitchell, D. W. (November 2007), "Solving cubics by solving triangles",
473:
found integer or rational solutions for some bivariate cubic equations (
15931:
15891:
15631:
15583:
15513:
15332:
15298:
15262:
15217:
15160:
14903:
14614:
14187:
14090:
11705:
of the roots instead of with the roots themselves. More precisely, let
6244:), then the other sign must be selected instead. If both choices yield
2428:{\displaystyle 18\,abcd-4\,b^{3}d+b^{2}c^{2}-4\,ac^{3}-27\,a^{2}d^{2}.}
1891:
482:
478:
470:
385:(fourth-degree) equations, but not for higher-degree equations, by the
14812:
9509:, the Galois group of most irreducible cubic polynomials is the group
4788:
and the other cube root by the other primitive cube root of the unity
3870:{\displaystyle ax^{3}+bx^{2}+cx+d=a\left(x+{\frac {b}{3a}}\right)^{3}}
350:, then it has at least one real root (this is true for all odd-degree
16061:
15379:(1869) , "Réflexions sur la résolution algébrique des équations", in
11418:
9093:
6840:
5916:
685:
434:
374:
15615:
15567:
15505:
15324:
15290:
15254:
15201:
15152:
11683:, but the computation is simpler with the depressed cubic equation,
8195:
Omar Khayyám's geometric solution of a cubic equation, for the case
7082:. It is purely real when the equation has three real roots (that is
16051:
14804:
14743:, vol. 2, Delhi, India: Bharattya Kala Prakashan, p. 76,
14423:
The Nine Chapters on the Mathematical Art: Companion and Commentary
14026:
14021:
13984:
9075:
1194:(The coefficients seem not to be integers, but must be integers if
242:
15003:
14649:(2nd ed.), New York: Chelsea Publishing Co., pp. 53–56,
15887:
15183:"A new approach to solving the cubic: Cardan's solution revealed"
14586:
Diophantus of Alexandria: A Study in the History of Greek Algebra
11571:
This implies that changing the sign of the square root exchanges
8282:
A simple modern proof is as follows. Multiplying the equation by
7259:{\displaystyle 4\cos ^{3}\theta -3\cos \theta -\cos(3\theta )=0.}
7151:
The formula can be proved as follows: Starting from the equation
4437:
of the equation is negative) then the equation has the real root
4133:{\displaystyle x_{1}={\frac {4abc-9a^{2}d-b^{3}}{a(b^{2}-3ac)}}.}
460:
358:
234:
37:
15550:
Dence, T. (November 1997), "Cubics, chaos and Newton's method",
11054:, and avoids the problem of computing two different cube roots.
5967:
times the resultant of the cubic and its second derivative, and
4720:
numbers, in this case. It was later shown (Cardano did not know
14737:; Singh, Avadhesh Narayan (2004), "Equation of Higher Degree",
14668:
A paper of Omar Khayyam, Scripta Math. 26 (1963), pages 323–337
9492:
if and only if the discriminant is the square of an element of
6836:
14357:. This change of sign allows getting directly the formulas of
9499:
As most integers are not squares, when working over the field
6816:
4781:{\displaystyle \varepsilon _{1}={\frac {-1+i{\sqrt {3}}}{2}},}
4269:
The above results are valid when the coefficients belong to a
15598:
Dunnett, R. (November 1994), "Newton–Raphson and the cubic",
8457:, cannot be solved by compass-and-straightedge construction.
327:
defined by the left-hand side of the equation. If all of the
14250:
This problem in turn led Khayyam to solve the cubic equation
12614:
are such symmetric polynomials (see below). It follows that
11818:
but this complex interpretation is not used here). Denoting
4312:. The formula applies to depressed cubics, but, as shown in
459:
text compiled around the 2nd century BC and commented on by
14017:
11301:{\displaystyle (w^{3})^{2}+q(w^{3})-{\frac {p^{3}}{27}}=0.}
10880:
the square root appearing in the formula is not real. As a
8484:
with three real roots, the roots are the projection on the
5436:{\displaystyle \textstyle {\frac {-1\pm {\sqrt {-3}}}{2}}.}
4254:{\displaystyle ax^{3}+bx^{2}+cx+d=a(x-x_{1})(x-x_{2})^{2}.}
4017:{\displaystyle x_{2}=x_{3}={\frac {9ad-bc}{2(b^{2}-3ac)}},}
2757:
is a root of a polynomial with real coefficients, then its
498:
15447:
Algebra in the Eighteenth Century: The Theory of Equations
14645:(1974) , "Chapter 8 Wang Hsiao-Tung and Cubic Equations",
13991:, because they are equivalent to solving a cubic equation.
12395:
is known to be zero in the case of a depressed cubic, and
9474:
is fixed by the Galois group only if the Galois group is
759:(1500–1557) received two problems in cubic equations from
14991:...if two roots are imaginary, the product is positive...
14197:
The speed of seismic Rayleigh waves is a solution of the
9252:
of all six permutations of the three roots, or the group
6811:
6704:
which means that the cubic polynomial can be factored as
5445:
This formula for the roots is always correct except when
932:, if the equation is reducible, one can suppose that the
15930:
15134:"Angle trisection, the heptagon, and the triskaidecagon"
14013:
of this cubic are the complex coordinates of those foci.
13686:, it is straightforward to express them in terms of the
11845:
the three roots of the cubic equation to be solved, let
11629:, in which case the only root of the depressed cubic is
8460:
7709:
3153:
If the discriminant of a cubic is zero, the cubic has a
15350:
A History of Algebra: From al-Khwārizmī to Emmy Noether
1865:
15755:(3rd ed.), New York: Cambridge University Press,
12998:
12837:
12297:
12206:
12128:
11742:
11531:
10767:
is a root of the equation; this is Cardano's formula.
9662:
and to substitute this in the depressed cubic, giving
6711:
5400:
4709:, below, for several methods for getting this result.
4426:{\displaystyle {\frac {q^{2}}{4}}+{\frac {p^{3}}{27}}}
1018:
14043:
13696:
13167:
13094:
13058:
13035:
12984:
12945:
12893:
12822:
12797:
12766:
12739:
12684:
12652:
12620:
12564:
12518:
12105:
11851:
11741:
11643:
Réflexions sur la résolution algébrique des équations
11530:
11425:, then the roots of the original depressed cubic are
11316:
11226:
11143:
10997:
10940:
10898:
10831:
10776:
10611:
10576:
10482:
10389:
10369:
10288:
10230:
10168:
9934:
9901:
9799:
9761:
9668:
9636:
9616:
9456:
9432:
9274:
8853:
8733:
8549:
8298:
7731:
7534:
7455:
7357:
7320:
7272:
7194:
7088:
6900:
6852:
6769:
6710:
6670:
6621:
6445:
6322:
6257:
6217:
6152:
6119:
6024:
5973:
5925:
5894:
5867:
5727:
5624:
5535:
5505:
5476:
5399:
5360:
5327:
5205:
5188:, the principal cube root is not defined in general.
5147:
5114:
5034:
4951:
4873:
4794:
4733:
4632:
4558:
4531:
4504:
4443:
4385:
4325:
4146:
4030:
3927:
3885:
3778:
3701:
3659:
3594:
3455:
3384:
3337:
3275:
3208:
3166:
3126:
3001:
2858:
2797:
2722:
2687:
2663:
2469:
2441:
2315:
2254:
2188:
2146:
2006:
1942:
1830:
1769:
1716:
1663:
1460:
1409:
1345:
1278:
1229:
1066:
1017:
945:
860:
714:
In the early 16th century, the Italian mathematician
602:
In the 11th century, the Persian poet-mathematician,
251:
65:
14621:. Translation by T. L. Heath. Rough Draft Printing.
14210:
of planetary motion is cubic in the semi-major axis.
11645:("Thoughts on the algebraic solving of equations"),
6756:{\displaystyle \textstyle a(x+{\frac {b}{3a}})^{3}.}
229:
and therefore the three real roots are 2, −1 and −4.
15387:, vol. III, Gauthier-Villars, pp. 205–421
14009:of the triangle's three vertices. The roots of the
12072:be the discrete Fourier transform of the roots. If
11653:, but Lagrange did not succeed in applying it to a
11390:be any nonzero root of this quadratic equation. If
11201:{\displaystyle w^{3}+q-{\frac {p^{3}}{27w^{3}}}=0.}
9925:one deduces that they are the two solutions of the
9558:This section regroups several methods for deriving
8541:, as shown above, the solution can be expressed as
3765:{\displaystyle x_{1}=x_{2}=x_{3}=-{\frac {b}{3a}},}
15753:Numerical Recipes: The Art of Scientific Computing
15460:Daniel Lazard, "Solving quintics in radicals", in
14494:
14060:
13828:
13666:
13143:A straightforward computation using the relations
13119:
13080:
13044:
13017:
12970:
12931:
12880:{\displaystyle x_{0}={\tfrac {1}{3}}(s_{1}+s_{2})}
12879:
12806:
12779:
12752:
12725:
12670:
12638:
12606:
12550:
12372:
12064:
11810:
11563:
11382:
11300:
11200:
11031:
10972:
10926:
10872:
10817:
10759:
10597:
10562:
10469:
10375:
10351:
10275:{\displaystyle \Delta =q^{2}+{\frac {4p^{3}}{27}}}
10274:
10216:
10154:
9917:
9879:
9785:
9745:
9654:
9622:
9466:
9442:
9387:
8975:-axis. Consequently, the roots of the equation in
8876:
8789:
8717:
8381:on the parabola. The equation of the circle being
8367:
8094:
7699:
7520:
7441:
7343:
7306:
7258:
7126:
7068:
6886:
6801:
6755:
6696:
6656:
6576:
6393:
6292:
6236:
6176:
6138:
6105:
5986:
5938:
5907:
5880:
5851:
5677:
5611:for the roots of the general cubic equation (with
5573:
5517:
5491:
5435:
5384:
5346:
5313:
5171:
5133:
5075:
5018:
4937:
4859:
4780:
4695:
4618:
4544:
4517:
4490:
4425:
4359:
4253:
4132:
4016:
3913:
3869:
3764:
3687:
3641:
3563:
3439:
3368:
3316:
3246:
3194:
3138:
3013:
2985:
2844:
2737:
2702:
2669:
2597:
2454:
2427:
2301:
2238:
2174:
2130:
1992:
1854:
1816:
1755:
1710:of the original equation are related to the roots
1702:
1644:
1446:
1375:
1331:
1257:
1186:
1048:
960:
916:
669:of a cubic equation. He also used the concepts of
304:
221:
15874:History of quadratic, cubic and quartic equations
15431:Mathematical Thought from Ancient to Modern Times
14979:. Longmans, Green, Reader, and Dyer. p. 13.
14824:
14762:
14647:The Development of Mathematics in China and Japan
14419:
14228:
14097:are the roots of a cubic polynomial which is the
13967:Cubic equations arise in various other contexts.
11564:{\displaystyle \textstyle -{\frac {p^{3}}{27W}}.}
7307:{\displaystyle u=2\,{\sqrt {-{\frac {p}{3}}}}\,,}
2239:{\displaystyle -\left(4\,p^{3}+27\,q^{2}\right).}
1870:The nature (real or not, distinct or not) of the
16145:
15397:
14529:Classical Algebra: Its Nature, Origins, and Uses
14286:More precisely, Vieta introduced a new variable
11032:{\displaystyle {\frac {-1\pm {\sqrt {-3}}}{2}}.}
9518:with six elements. An example of a Galois group
8425:
7721:), this root can be similarly represented using
7188:to make the equation coincide with the identity
1049:{\displaystyle \textstyle x_{1}={\frac {p}{q}},}
764:money. Tartaglia received questions in the form
609:Treatise on Demonstration of Problems of Algebra
365:involving the four coefficients, the four basic
48:(where the curve crosses the horizontal axis at
15344:
15092:https://mathworld.wolfram.com/CubicFormula.html
11046:Vieta's substitution is a method introduced by
10217:{\displaystyle x^{2}+qx-{\frac {p^{3}}{27}}=0.}
9099:is the abscissa of the tangency point H of the
8213:. The intersection of the vertical line on the
992:by examining a finite number of cases (because
15953:Zero polynomial (degree undefined or −1 or −∞)
11524:. The other root of the quadratic equation is
4491:{\displaystyle {\sqrt{u_{1}}}+{\sqrt{u_{2}}},}
3440:{\displaystyle t_{2}=t_{3}=-{\frac {3q}{2p}}.}
497:, but also discussed the conditions where the
15916:
14710:Episodes in the Mathematics of Medieval Islam
10892:(1572). The solution is to use the fact that
9746:{\displaystyle u^{3}+v^{3}+(3uv+p)(u+v)+q=0.}
9399:is the leading coefficient of the cubic, and
7449:Combining with the above identity, one gets
6664:the formula gives that the three roots equal
5199:. It results that a root of the equation is
4867:That is, the other roots of the equation are
2745:the cubic has one real root and two non-real
2614:To prove the preceding formulas, one can use
361:: more precisely, they can be expressed by a
14883:
13987:that have been proved to not be solvable by
13682:are symmetric functions of the roots. Using
12816:In the case of the depressed cubic, one has
12678:are the two roots of the quadratic equation
9755:At this point Cardano imposed the condition
9450:changes of sign if two roots are exchanged,
9074:, if there is only one real root, it is the
8445:root. This implies that the old problems of
6566:
6548:
4706:
1817:{\displaystyle x_{i}=t_{i}-{\frac {b}{3a}},}
1763:of the depressed equation by the relations
1339:be a cubic equation. The change of variable
1207:Then, the other roots are the roots of this
846:If the coefficients of a cubic equation are
15749:"Section 5.6 Quadratic and Cubic Equations"
15310:
15118:Abramowitz, Milton; Stegun, Irene A., eds.
15094:, rewritten for having a coherent notation.
14924:
14922:
14920:
14740:History of Hindu Mathematics: A Source Book
14677:J. J. O'Connor and E. F. Robertson (1999),
14492:
14420:Crossley, John; W.-C. Lun, Anthony (1999).
14322:. This is equivalent with the substitution
10359:So (without loss of generality in choosing
6817:Trigonometric solution for three real roots
5470:. However, Cardano's formula is useless if
3050:are complex conjugates, which implies that
15923:
15909:
14931:"Viète, Descartes, and the cubic equation"
14733:
14613:
14563:
14561:
14358:
14108:of a third-order constant coefficients or
12476:), but some simple symmetric functions of
11701:Lagrange's main idea was to work with the
10981:
9559:
9553:
8877:{\displaystyle 2{\sqrt {-{\frac {p}{3}}}}}
8504:. The center of the triangle has the same
8186:
6802:{\displaystyle \Delta _{0}=\Delta _{1}=0.}
6657:{\displaystyle \Delta _{0}=\Delta _{1}=0,}
6293:{\displaystyle \Delta _{0}=\Delta _{1}=0,}
3269:, and 0 is a triple root of the cubic. If
319:The solutions of this equation are called
15401:Elliptic functions and elliptic integrals
15398:Prasolov, Viktor; Solovyev, Yuri (1997),
15238:
15004:"Solution for a depressed cubic equation"
13131:
11051:
9606:. The idea is to introduce two variables
9591:This method applies to a depressed cubic
8987:
8711:
7569:
7551:
7514:
7415:
7300:
7282:
7033:
7006:
6947:
6935:
6917:
4316:, it allows solving all cubic equations.
4264:
3088:is real and negative. On the other hand,
2753:This can be proved as follows. First, if
2401:
2381:
2338:
2319:
2217:
2200:
924:with integer coefficients, is said to be
854:of their denominators. Such an equation
452:The Nine Chapters on the Mathematical Art
15491:
15375:
15180:
14928:
14917:
14877:
14793:Journal of the American Oriental Society
14790:
14706:
14694:
14567:
14552:
14426:. Oxford University Press. p. 176.
14112:(equidimensional variable coefficients)
8991:
8469:
8438:if and only if it has three real roots.
8263:, the circle that has as a diameter the
8190:
5602:
4713:
3160:The discriminant of the depressed cubic
2653:If the coefficients of a polynomial are
2618:to express everything as polynomials in
2140:The discriminant of the depressed cubic
838:(1596–1650) extended the work of Viète.
746:
503:
36:
15527:Anglin, W. S.; Lambek, Joachim (1995),
15131:
15125:
15063:
14836:MacTutor History of Mathematics Archive
14774:MacTutor History of Mathematics Archive
14683:MacTutor History of Mathematics archive
14558:
14240:MacTutor History of Mathematics Archive
14169:, can be solved using a cubic equation.
11041:
9117:
8797:is an angle in the unit circle; taking
6439:. In other words, the three roots are
824:, but he did not really understand it.
16146:
15456:
15454:
15026:
14641:
14501:. Greenwood Publishing Group. p.
14380:
14165:, which can be used to find the pH of
13872:in the case of a depressed cubic, and
12939:while in Cardano's method we have set
9483:. In other words, the Galois group is
8416:, the right hand side is the value of
8181:
7714:When there is only one real root (and
6812:Trigonometric and hyperbolic solutions
2648:
2248:The discriminant of the general cubic
559:systematically established and solved
15904:
15653:, vol. 1 (2nd ed.), Dover,
15427:
15132:Gleason, Andrew Mattei (March 1988).
14972:
14579:
14532:. John Wiley & Sons. p. 64.
14525:
14472:. John Wiley & Sons. p. 63.
14465:
14445:
14443:
14415:
14413:
13989:straightedge and compass construction
13018:{\displaystyle uv=-{\tfrac {1}{3}}p.}
12607:{\displaystyle S=s_{1}^{3}+s_{2}^{3}}
10224:The discriminant of this equation is
9426:are the three roots of the cubic. As
8461:Geometric interpretation of the roots
7710:Hyperbolic solution for one real root
3369:{\displaystyle t_{1}={\frac {3q}{p}}}
917:{\displaystyle ax^{3}+bx^{2}+cx+d=0,}
447:compass and straightedge construction
15391:
15274:
15181:Nickalls, R. W. D. (November 1993),
14852:
14151:
11636:
11137:transforms the depressed cubic into
10873:{\displaystyle 4p^{3}+27q^{2}<0,}
10818:{\displaystyle 4p^{3}+27q^{2}>0,}
9261:of the three circular permutations.
7145:
6844:
6196:" before the square root is either "
5686:
5678:{\displaystyle ax^{3}+bx^{2}+cx+d=0}
5463:, the square root is chosen so that
5076:{\displaystyle 4p^{3}+27q^{2}<0,}
4313:
4296:
3585:
1866:Discriminant and nature of the roots
1332:{\displaystyle ax^{3}+bx^{2}+cx+d=0}
555:in his mathematical treatise titled
305:{\displaystyle ax^{3}+bx^{2}+cx+d=0}
15451:
15348:(1985), "From Viète to Descartes",
15057:
14769:"Sharaf al-Din al-Muzaffar al-Tusi"
11216:, one gets a quadratic equation in
10973:{\displaystyle v={\frac {-p}{3u}}.}
10927:{\displaystyle uv=-{\frac {p}{3}},}
9889:Knowing the sum and the product of
9582:who first published it in his book
9565:
8465:
8375:The left-hand side is the value of
7344:{\displaystyle {\frac {u^{3}}{4}}.}
7127:{\displaystyle 4p^{3}+27q^{2}<0}
6843:. More precisely, the roots of the
5499:as the roots are the cube roots of
4716:, the two other roots are non-real
4433:is positive (this implies that the
3331:, then the cubic has a simple root
1993:{\displaystyle ax^{3}+bx^{2}+cx+d,}
1376:{\displaystyle x=t-{\frac {b}{3a}}}
13:
15520:
15472:, pp. 207–225, Berlin, 2004.
14859:. Boston: Addison Wesley. p.
14486:
14440:
14410:
12436:need to be computed. They are not
11057:Starting from the depressed cubic
10231:
9459:
9435:
9277:
6784:
6771:
6636:
6623:
6507:
6407:, or, equivalently by multiplying
6367:
6272:
6259:
6219:
6072:
6051:
6036:
5975:
5927:
5919:of the cubic and its derivatives:
5896:
5869:
5779:
5733:
5583:This formula is also correct when
3642:{\displaystyle ax^{3}+bx^{2}+cx+d}
3127:
3002:
2859:
2845:{\displaystyle ax^{3}+bx^{2}+cx+d}
2765:, at least one root must be real.
2723:
2710:the cubic has three distinct real
2688:
2677:is not zero, there are two cases:
2664:
2302:{\displaystyle ax^{3}+bx^{2}+cx+d}
1218:
795:Later, Tartaglia was persuaded by
684:, Leonardo de Pisa, also known as
14:
16175:
15849:
15141:The American Mathematical Monthly
14519:
14497:Daily Life in Ancient Mesopotamia
13970:
12726:{\displaystyle z^{2}-Sz+P^{3}=0.}
12512:In the case of a cubic equation,
9467:{\displaystyle {\sqrt {\Delta }}}
9443:{\displaystyle {\sqrt {\Delta }}}
6697:{\displaystyle {\frac {-b}{3a}},}
6139:{\displaystyle {\sqrt {{~}^{~}}}}
5574:{\displaystyle 4p^{3}+27q^{2}=0.}
5347:{\displaystyle {\sqrt {{~}^{~}}}}
5134:{\displaystyle {\sqrt {{~}^{~}}}}
3317:{\displaystyle 4p^{3}+27q^{2}=0,}
3247:{\displaystyle 4p^{3}+27q^{2}=0.}
2791:are the three roots of the cubic
1756:{\displaystyle t_{1},t_{2},t_{3}}
1703:{\displaystyle x_{1},x_{2},x_{3}}
41:Graph of a cubic function with 3
15529:"Mathematics in the Renaissance"
15067:CRC Standard Mathematical Tables
14493:Nemet-Nejat, Karen Rhea (1998).
14466:Cooke, Roger (8 November 2012).
12418:for the general cubic. So, only
9527:with three elements is given by
8982:
8292:and regrouping the terms gives
6177:{\displaystyle {\sqrt{{~}^{~}}}}
5385:{\displaystyle {\sqrt{{~}^{~}}}}
5172:{\displaystyle {\sqrt{{~}^{~}}}}
5083:there are three real roots, but
3148:
841:
400:of the roots can be found using
15535:, Springers, pp. 125–131,
15485:
15470:The Legacy of Niels Henrik Abel
15421:
15369:
15346:van der Waerden, Bartel Leenert
15338:
15304:
15268:
15232:
15174:
15122:, Dover (1965), chap. 22 p. 773
15112:
15097:
15084:
15020:
14996:
14966:
14846:
14818:
14784:
14756:
14727:
14700:
14688:
14671:
14662:
14635:
14607:
14280:
14221:
14182:Van der Waals equation of state
14075:is one of the roots of a cubic.
13962:
12932:{\displaystyle s_{1}s_{2}=-3p,}
9268:of the cubic is the square of
9202:
8689:
8683:
8221:For solving the cubic equation
8070:
7890:
7670:
7039:
6541:
5233:
5227:
1881:
639:. In the 12th century, another
563:25 cubic equations of the form
16:Polynomial equation of degree 3
15434:, Oxford University Press US,
14929:Nickalls, R.W.D. (July 2006).
14891:The Mathematical Intelligencer
14707:Berggren, J. L. (2017-01-18).
14573:
14546:
14459:
14374:
14082:relies on the solution of its
13688:elementary symmetric functions
13618:
13450:
13441:
13387:
13328:
13259:
12874:
12848:
12360:
12308:
12269:
12217:
12178:
12139:
11731:(when working in the space of
11269:
11256:
11241:
11227:
10139:
10129:
10120:
10094:
10042:
10016:
9990:
9971:
9968:
9949:
9728:
9716:
9713:
9695:
9379:
9353:
9350:
9324:
9321:
9295:
7848:
7840:
7771:
7763:
7471:
7462:
7247:
7238:
6740:
6715:
4707:§ Derivation of the roots
4367:is a cubic equation such that
4239:
4219:
4216:
4197:
4121:
4096:
4005:
3980:
3914:{\displaystyle b^{2}\neq 3ac,}
3586:reduction of a depressed cubic
2974:
2947:
2938:
2911:
2902:
2875:
2763:fundamental theorem of algebra
2565:
2514:
2502:
2476:
2116:
2089:
2080:
2053:
2044:
2017:
1932:(not necessarily distinct nor
1269:to that of a depressed cubic.
1211:and can be found by using the
212:
200:
197:
185:
182:
170:
79:
73:
1:
16113:Horner's method of evaluation
15278:American Mathematical Monthly
15242:American Mathematical Monthly
15064:Shelbey, Samuel, ed. (1975).
14368:
14123:Intersection points of cubic
12551:{\displaystyle P=s_{1}s_{2},}
11713:, that is a number such that
11711:primitive third root of unity
10989:primitive cube roots of unity
9235:of the smallest extension of
9103:to cubic that passes through
8426:Solution with angle trisector
6237:{\displaystyle \Delta _{0}=0}
5321:In this formula, the symbols
4319:Cardano's result is that if
3139:{\displaystyle \Delta <0.}
3014:{\displaystyle \Delta >0.}
2738:{\displaystyle \Delta <0,}
2703:{\displaystyle \Delta >0,}
1447:{\displaystyle t^{3}+pt+q=0,}
14888:(2002), "Reading Bombelli",
14395:10.1007/978-3-0348-8599-7_16
14114:linear differential equation
14078:The solution of the general
13983:are two ancient problems of
13025:Thus, up to the exchange of
9219:different from 2 and 3, the
8455:ancient Greek mathematicians
6887:{\displaystyle t^{3}+pt+q=0}
6413:primitive cube root of unity
4726:primitive cube root of unity
4360:{\displaystyle t^{3}+pt+q=0}
3921:the cubic has a double root
3695:the cubic has a triple root
2852:, then the discriminant is
539: − 1) (
7:
16118:Polynomial identity testing
15862:Encyclopedia of Mathematics
15109:86. November 2002, 473–477.
8891:
8887:For the non-depressed case
8520:
8478:
7314:and divide the equation by
6835:, specifically in terms of
5987:{\displaystyle \Delta _{0}}
5939:{\displaystyle \Delta _{1}}
5908:{\displaystyle \Delta _{1}}
5881:{\displaystyle \Delta _{0}}
5456:, with the proviso that if
936:have integer coefficients.
466:In the 3rd century AD, the
10:
16180:
14469:The History of Mathematics
14290:and imposed the condition
14184:, are cubic in the volume.
11703:discrete Fourier transform
11072:, Vieta's substitution is
4714:§ Nature of the roots
3688:{\displaystyle b^{2}=3ac,}
3195:{\displaystyle t^{3}+pt+q}
2175:{\displaystyle t^{3}+pt+q}
1258:{\displaystyle t^{3}+pt+q}
428:
346:of the cubic equation are
55:). The case shown has two
29:
18:
16090:
16029:
15942:
15827:10.1017/S0025557200183135
15787:10.1017/S0025557200183147
15725:10.1017/S0025557200185237
15685:10.1017/S0025557200182178
15043:10.1017/S0025557200183135
14952:10.1017/S0025557200179598
14099:characteristic polynomial
13120:{\displaystyle s_{2}=3v.}
12971:{\displaystyle x_{0}=u+v}
12671:{\displaystyle s_{2}^{3}}
12639:{\displaystyle s_{1}^{3}}
12440:of the roots (exchanging
8249:constructed the parabola
8128:are sometimes called the
5529:; this includes the case
4310:Niccolo Fontana Tartaglia
2000:then the discriminant is
751:Niccolò Fontana Tartaglia
551:astronomer mathematician
15819:Mathematical Association
15779:Mathematical Association
15717:Mathematical Association
15677:Mathematical Association
15608:Mathematical Association
15560:Mathematical Association
15190:The Mathematical Gazette
14856:A History of Mathematics
14841:University of St Andrews
14779:University of St Andrews
14589:. Martino Pub. pp.
14359:§ Cardano's formula
14245:University of St Andrews
14214:
14068:satisfy cubic equations.
13081:{\displaystyle s_{1}=3u}
10982:§ Cardano's formula
9786:{\displaystyle 3uv+p=0.}
9546:, whose discriminant is
8279:-axis (see the figure).
7528:and the roots are thus
1402:depressed cubic equation
1060:. This other factor is
1058:polynomial long division
547:In the 7th century, the
398:numerical approximations
377:. (This is also true of
30:Not to be confused with
16103:Greatest common divisor
15494:Mathematics News Letter
14619:The works of Archimedes
14106:characteristic equation
14073:one-third of that angle
14061:{\displaystyle 2\pi /7}
13959:, in the general case.
11052:§ Cardano's method
9554:Derivation of the roots
8187:Omar Khayyám's solution
7078:This formula is due to
6833:trigonometric functions
3068:purely imaginary number
2670:{\displaystyle \Delta }
2657:, and its discriminant
1855:{\displaystyle i=1,2,3}
663:numerically approximate
647:(1135–1213), wrote the
592:, and two of them with
535: + 1) (2
402:root-finding algorithms
59:. Here the function is
15975:Quadratic function (2)
15533:The Heritage of Thales
15428:Kline, Morris (1990),
15377:Lagrange, Joseph-Louis
14062:
13830:
13668:
13121:
13082:
13046:
13019:
12972:
12933:
12881:
12808:
12781:
12754:
12727:
12672:
12640:
12608:
12552:
12374:
12066:
11812:
11565:
11384:
11302:
11202:
11033:
10974:
10928:
10874:
10819:
10761:
10599:
10598:{\displaystyle u+v=t,}
10564:
10471:
10377:
10353:
10276:
10218:
10156:
9919:
9918:{\displaystyle v^{3},}
9881:
9787:
9747:
9656:
9624:
9570:This method is due to
9468:
9444:
9389:
9209:irreducible polynomial
9063:
8988:In the Cartesian plane
8878:
8791:
8719:
8513:
8488:-axis of the vertices
8369:
8218:
8119:, the above values of
8096:
7701:
7522:
7443:
7345:
7308:
7260:
7184:The idea is to choose
7146:§ Depressed cubic
7128:
7070:
6888:
6803:
6757:
6698:
6658:
6578:
6395:
6294:
6238:
6178:
6140:
6107:
5988:
5940:
5909:
5882:
5853:
5679:
5575:
5519:
5493:
5437:
5386:
5348:
5315:
5173:
5135:
5077:
5020:
4939:
4861:
4782:
4697:
4620:
4546:
4519:
4492:
4427:
4361:
4314:§ Depressed cubic
4265:Characteristic 2 and 3
4255:
4134:
4018:
3915:
3871:
3766:
3689:
3643:
3565:
3441:
3370:
3318:
3248:
3196:
3140:
3023:If only one root, say
3015:
2987:
2846:
2739:
2704:
2671:
2599:
2456:
2429:
2303:
2240:
2176:
2132:
1994:
1856:
1818:
1757:
1704:
1646:
1448:
1387:) that has no term in
1377:
1333:
1259:
1188:
1050:
962:
918:
752:
622:
544:
511:of the cubic function
306:
241:in one variable is an
230:
223:
15958:Constant function (0)
15381:Serret, Joseph-Alfred
15313:Annals of Mathematics
14973:Pratt, Orson (1866).
14853:Katz, Victor (2004).
14735:Datta, Bibhutibhushan
14526:Cooke, Roger (2008).
14192:rates of acceleration
14063:
13831:
13690:of the roots, giving
13669:
13122:
13083:
13047:
13020:
12973:
12934:
12882:
12809:
12782:
12780:{\displaystyle s_{2}}
12755:
12753:{\displaystyle s_{1}}
12728:
12673:
12641:
12609:
12553:
12375:
12067:
11813:
11647:Joseph Louis Lagrange
11566:
11385:
11303:
11203:
11034:
10975:
10929:
10875:
10820:
10770:This works well when
10762:
10600:
10565:
10472:
10378:
10354:
10277:
10219:
10157:
9920:
9882:
9788:
9748:
9657:
9655:{\displaystyle u+v=t}
9625:
9578:, but is named after
9469:
9445:
9390:
8995:
8879:
8792:
8720:
8473:
8432:compass, straightedge
8370:
8194:
8097:
7702:
7523:
7444:
7346:
7309:
7261:
7129:
7071:
6889:
6804:
6758:
6699:
6659:
6579:
6396:
6295:
6239:
6179:
6141:
6108:
5989:
5941:
5910:
5883:
5854:
5680:
5603:General cubic formula
5576:
5520:
5494:
5438:
5387:
5349:
5316:
5174:
5136:
5078:
5021:
4940:
4862:
4783:
4698:
4621:
4547:
4545:{\displaystyle u_{2}}
4520:
4518:{\displaystyle u_{1}}
4493:
4428:
4362:
4256:
4135:
4019:
3916:
3872:
3767:
3690:
3644:
3566:
3442:
3371:
3319:
3249:
3197:
3141:
3016:
2988:
2847:
2740:
2705:
2672:
2600:
2457:
2455:{\displaystyle a^{4}}
2435:It is the product of
2430:
2304:
2241:
2177:
2133:
1995:
1857:
1819:
1758:
1705:
1647:
1449:
1378:
1334:
1260:
1189:
1051:
1004:must be a divisor of
996:must be a divisor of
963:
919:
750:
658:Horner–Ruffini method
653:Treatise on Equations
645:Sharaf al-Dīn al-Tūsī
618:
543: − 2)
507:
475:Diophantine equations
463:in the 3rd century.
367:arithmetic operations
307:
224:
40:
16091:Tools and algorithms
16011:Quintic function (5)
15999:Quartic function (4)
15936:polynomial functions
15811:Mathematical Gazette
15771:Mathematical Gazette
15709:Mathematical Gazette
15669:Mathematical Gazette
15600:Mathematical Gazette
15552:Mathematical Gazette
15106:Mathematical Gazette
15030:Mathematical Gazette
14939:Mathematical Gazette
14827:Robertson, Edmund F.
14765:Robertson, Edmund F.
14389:, pp. 315–358,
14271:trigonometric tables
14231:Robertson, Edmund F.
14159:analytical chemistry
14120:is a cubic equation.
14041:
13694:
13165:
13092:
13056:
13033:
12982:
12943:
12891:
12820:
12795:
12764:
12737:
12682:
12650:
12618:
12562:
12516:
12103:
11849:
11739:
11659:Abel–Ruffini theorem
11528:
11314:
11224:
11141:
11042:Vieta's substitution
10995:
10938:
10896:
10829:
10774:
10609:
10574:
10480:
10387:
10367:
10286:
10228:
10166:
9932:
9899:
9797:
9759:
9666:
9634:
9614:
9454:
9430:
9272:
9227:is the group of the
9118:In the complex plane
9066:When the graph of a
9018:(negative here) and
8884:corrects for scale.
8851:
8731:
8547:
8502:equilateral triangle
8296:
8130:Chebyshev cube root.
7729:
7723:hyperbolic functions
7532:
7453:
7355:
7318:
7270:
7192:
7086:
6898:
6850:
6767:
6708:
6668:
6619:
6443:
6320:
6255:
6215:
6150:
6117:
6022:
5971:
5923:
5915:can be expressed as
5892:
5865:
5725:
5622:
5533:
5503:
5492:{\displaystyle p=0,}
5474:
5397:
5358:
5325:
5203:
5179:as representing the
5145:
5112:
5089:algebraic expression
5032:
4949:
4871:
4792:
4731:
4630:
4556:
4552:are the two numbers
4529:
4502:
4441:
4383:
4323:
4144:
4028:
3925:
3883:
3776:
3699:
3657:
3592:
3453:
3382:
3335:
3273:
3206:
3164:
3124:
2999:
2856:
2795:
2768:As stated above, if
2720:
2685:
2661:
2467:
2439:
2313:
2252:
2186:
2144:
2004:
1940:
1828:
1767:
1714:
1661:
1458:
1407:
1343:
1276:
1227:
1223:Cubics of the form
1209:quadratic polynomial
1064:
1015:
961:{\displaystyle qx-p}
943:
858:
740:to be negative, but
531: + 2 = (
527: − 3
523: − 3
457:Chinese mathematical
387:Abel–Ruffini theorem
381:(second-degree) and
352:polynomial functions
249:
63:
16021:Septic equation (7)
16016:Sextic equation (6)
15963:Linear function (1)
15462:Olav Arnfinn Laudal
14884:La Nave, Federica;
14825:O'Connor, John J.;
14763:O'Connor, John J.;
14617:(October 8, 2007).
14229:O'Connor, John J.;
14188:Kinematic equations
14118:difference equation
14035:heptagonal triangle
13776:
13725:
13684:Newton's identities
13617:
13589:
13561:
13523:
13495:
13467:
13440:
13422:
13404:
13380:
13362:
13255:
13237:
13219:
12667:
12635:
12603:
12585:
12438:symmetric functions
9229:field automorphisms
8508:-coordinate as the
8182:Geometric solutions
6828:casus irreducibilis
6184:are interpreted as
6085:
6064:
5599:other than 2 or 3.
5518:{\displaystyle -q.}
5106:casus irreducibilis
5094:casus irreducibilis
4822:
4024:and a simple root,
3575:system of equations
3258:is also zero, then
2649:Nature of the roots
705:Babylonian numerals
468:Greek mathematician
16154:Elementary algebra
15987:Cubic function (3)
15980:Quadratic equation
15385:Œuvres de Lagrange
14904:10.1007/BF03025306
14583:(April 30, 2009).
14208:Kepler's third law
14178:equations of state
14058:
13826:
13824:
13762:
13711:
13664:
13662:
13603:
13575:
13547:
13509:
13481:
13453:
13426:
13408:
13390:
13366:
13348:
13241:
13223:
13205:
13117:
13078:
13045:{\displaystyle v,}
13042:
13015:
13007:
12968:
12929:
12877:
12846:
12807:{\displaystyle v.}
12804:
12777:
12750:
12723:
12668:
12653:
12636:
12621:
12604:
12589:
12571:
12548:
12370:
12368:
12306:
12215:
12137:
12062:
12060:
11808:
11807:
11561:
11560:
11380:
11298:
11198:
11029:
10970:
10924:
10870:
10815:
10757:
10595:
10560:
10467:
10373:
10349:
10272:
10214:
10152:
10150:
9927:quadratic equation
9915:
9877:
9875:
9783:
9743:
9652:
9620:
9572:Scipione del Ferro
9464:
9440:
9385:
9070:is plotted in the
9064:
8874:
8787:
8715:
8514:
8365:
8219:
8209:, giving the root
8092:
8090:
7697:
7518:
7439:
7341:
7304:
7256:
7124:
7066:
6884:
6799:
6753:
6752:
6694:
6654:
6574:
6391:
6290:
6234:
6174:
6136:
6113:where the symbols
6103:
6071:
6050:
5984:
5936:
5905:
5878:
5849:
5847:
5675:
5571:
5515:
5489:
5433:
5432:
5382:
5344:
5311:
5169:
5131:
5073:
5016:
4935:
4857:
4808:
4778:
4693:
4616:
4542:
4515:
4488:
4423:
4357:
4306:Scipione del Ferro
4251:
4130:
4014:
3911:
3867:
3762:
3685:
3639:
3561:
3437:
3378:and a double root
3366:
3314:
3244:
3192:
3136:
3011:
2983:
2842:
2735:
2700:
2667:
2595:
2452:
2425:
2299:
2236:
2172:
2128:
1990:
1852:
1814:
1753:
1700:
1642:
1640:
1444:
1396:After dividing by
1383:gives a cubic (in
1373:
1329:
1267:change of variable
1255:
1184:
1046:
1045:
1011:Thus, one root is
982:rational root test
958:
914:
753:
716:Scipione del Ferro
581:, 23 of them with
545:
302:
231:
219:
217:
16141:
16140:
16082:Quasi-homogeneous
15896:quartic equations
15857:"Cardano formula"
15762:978-0-521-88068-8
15660:978-0-486-47189-1
15542:978-0-387-94544-6
15441:978-0-19-506136-9
15411:978-0-8218-0587-9
15404:, AMS Bookstore,
14720:978-1-4939-3780-6
14656:978-0-8284-0149-4
14539:978-0-470-27797-3
14512:978-0-313-29497-6
14479:978-1-118-46029-0
14433:978-0-19-853936-0
14404:978-3-0348-8599-7
14190:involving linear
14152:In other sciences
14141:Inflection points
14003:Steiner inellipse
13981:doubling the cube
13006:
12845:
12305:
12214:
12136:
11775:
11769:
11651:quartic equations
11637:Lagrange's method
11555:
11378:
11376:
11356:
11334:
11290:
11190:
11106:The substitution
11024:
11018:
10965:
10919:
10755:
10747:
10745:
10725:
10703:
10684:
10676:
10674:
10654:
10632:
10555:
10547:
10545:
10525:
10503:
10462:
10454:
10452:
10432:
10410:
10376:{\displaystyle v}
10344:
10342:
10322:
10300:
10270:
10206:
9868:
9623:{\displaystyle v}
9560:Cardano's formula
9462:
9438:
9280:
9264:The discriminant
9137:Steiner inellipse
8872:
8870:
8780:
8779:
8763:
8687:
8676:
8650:
8649:
8633:
8602:
8581:
8579:
8451:doubling the cube
8349:
8321:
8141:analytic function
8078:
8074:
8058:
8057:
8046:
8015:
7994:
7993:
7944:
7940:
7936:
7898:
7894:
7878:
7877:
7861:
7817:
7796:
7794:
7779:
7674:
7663:
7637:
7636:
7620:
7589:
7567:
7565:
7512:
7511:
7495:
7431:
7430:
7413:
7336:
7298:
7296:
7266:For this, choose
7043:
7031:
7004:
7003:
6987:
6956:
6933:
6931:
6737:
6689:
6572:
6531:
6475:
6389:
6379:
6345:
6172:
6164:
6159:
6134:
6131:
6126:
6098:
6092:
6086:
5707:belonging to any
5427:
5421:
5380:
5372:
5367:
5342:
5339:
5334:
5306:
5298:
5296:
5276:
5254:
5231:
5225:
5167:
5159:
5154:
5129:
5126:
5121:
5011:
4979:
4933:
4901:
4852:
4846:
4773:
4767:
4718:complex conjugate
4688:
4686:
4666:
4644:
4614:
4612:
4592:
4570:
4483:
4461:
4421:
4401:
4297:Cardano's formula
4283:formal derivative
4125:
4009:
3854:
3757:
3545:
3508:
3432:
3364:
2759:complex conjugate
2747:complex conjugate
2590:
1809:
1633:
1551:
1494:
1371:
1213:quadratic formula
1182:
1119:
1075:
1040:
757:Niccolò Tartaglia
671:maxima and minima
439:doubling the cube
393:trigonometrically
168:
97:
25:cubic plane curve
16171:
16004:Quartic equation
15925:
15918:
15911:
15902:
15901:
15878:MacTutor archive
15870:
15845:
15805:
15765:
15743:
15703:
15663:
15647:Jacobson, Nathan
15642:
15594:
15545:
15516:
15480:
15458:
15449:
15444:
15425:
15419:
15414:
15395:
15389:
15388:
15373:
15367:
15366:
15342:
15336:
15335:
15308:
15302:
15301:
15272:
15266:
15265:
15236:
15230:
15229:See esp. Fig. 2.
15228:
15196:(480): 354–359,
15187:
15178:
15172:
15171:
15169:
15163:. Archived from
15138:
15129:
15123:
15116:
15110:
15101:
15095:
15088:
15082:
15081:
15061:
15055:
15054:
15024:
15018:
15017:
15015:
15014:
15000:
14994:
14993:
14970:
14964:
14963:
14946:(518): 203–208.
14935:
14926:
14915:
14914:
14881:
14875:
14874:
14850:
14844:
14843:
14822:
14816:
14815:
14788:
14782:
14781:
14760:
14754:
14753:
14731:
14725:
14724:
14704:
14698:
14692:
14686:
14675:
14669:
14666:
14660:
14659:
14639:
14633:
14632:
14611:
14605:
14604:
14581:Heath, Thomas L.
14577:
14571:
14565:
14556:
14550:
14544:
14543:
14523:
14517:
14516:
14500:
14490:
14484:
14483:
14463:
14457:
14447:
14438:
14437:
14417:
14408:
14407:
14378:
14362:
14356:
14352:
14347:
14345:
14344:
14338:
14335:
14321:
14320:
14318:
14317:
14314:
14311:
14289:
14284:
14278:
14265:
14247:
14225:
14167:buffer solutions
14163:Charlot equation
14145:quintic function
14135:quartic function
14080:quartic equation
14067:
14065:
14064:
14059:
14054:
14011:first derivative
13997:states that the
13995:Marden's theorem
13977:Angle trisection
13958:
13957:
13955:
13954:
13949:
13946:
13929:
13928:
13926:
13925:
13920:
13917:
13900:
13899:
13897:
13896:
13891:
13888:
13871:
13858:
13845:
13835:
13833:
13832:
13827:
13825:
13818:
13817:
13802:
13801:
13792:
13791:
13775:
13770:
13741:
13740:
13724:
13719:
13681:
13677:
13674:This shows that
13673:
13671:
13670:
13665:
13663:
13656:
13655:
13646:
13645:
13636:
13635:
13616:
13611:
13602:
13601:
13588:
13583:
13574:
13573:
13560:
13555:
13546:
13545:
13533:
13532:
13522:
13517:
13505:
13504:
13494:
13489:
13477:
13476:
13466:
13461:
13439:
13434:
13421:
13416:
13403:
13398:
13379:
13374:
13361:
13356:
13327:
13326:
13317:
13316:
13304:
13303:
13294:
13293:
13281:
13280:
13271:
13270:
13254:
13249:
13236:
13231:
13218:
13213:
13201:
13200:
13191:
13190:
13160:
13149:
13139:
13135:
13126:
13124:
13123:
13118:
13104:
13103:
13087:
13085:
13084:
13079:
13068:
13067:
13051:
13049:
13048:
13043:
13028:
13024:
13022:
13021:
13016:
13008:
12999:
12977:
12975:
12974:
12969:
12955:
12954:
12938:
12936:
12935:
12930:
12913:
12912:
12903:
12902:
12886:
12884:
12883:
12878:
12873:
12872:
12860:
12859:
12847:
12838:
12832:
12831:
12813:
12811:
12810:
12805:
12790:
12786:
12784:
12783:
12778:
12776:
12775:
12759:
12757:
12756:
12751:
12749:
12748:
12732:
12730:
12729:
12724:
12716:
12715:
12694:
12693:
12677:
12675:
12674:
12669:
12666:
12661:
12645:
12643:
12642:
12637:
12634:
12629:
12613:
12611:
12610:
12605:
12602:
12597:
12584:
12579:
12557:
12555:
12554:
12549:
12544:
12543:
12534:
12533:
12508:
12501:
12493:
12484:
12475:
12466:
12457:
12448:
12435:
12426:
12417:
12416:
12414:
12413:
12408:
12405:
12394:
12384:Vieta's formulas
12379:
12377:
12376:
12371:
12369:
12359:
12358:
12349:
12348:
12336:
12335:
12320:
12319:
12307:
12298:
12288:
12287:
12268:
12267:
12252:
12251:
12242:
12241:
12229:
12228:
12216:
12207:
12197:
12196:
12177:
12176:
12164:
12163:
12151:
12150:
12138:
12129:
12119:
12118:
12098:
12089:
12080:
12071:
12069:
12068:
12063:
12061:
12054:
12053:
12038:
12037:
12028:
12027:
12015:
12014:
11998:
11997:
11981:
11980:
11971:
11970:
11958:
11957:
11942:
11941:
11925:
11924:
11908:
11907:
11895:
11894:
11882:
11881:
11865:
11864:
11844:
11835:
11826:
11817:
11815:
11814:
11809:
11803:
11802:
11798:
11776:
11771:
11770:
11765:
11750:
11730:
11719:
11708:
11697:
11682:
11655:quintic equation
11632:
11628:
11617:
11610:
11609:
11607:
11606:
11595:
11592:
11581:
11570:
11568:
11567:
11562:
11556:
11554:
11546:
11545:
11536:
11523:
11522:
11520:
11519:
11510:
11507:
11490:
11489:
11487:
11486:
11477:
11474:
11457:
11456:
11454:
11453:
11444:
11441:
11424:
11416:
11407:
11398:
11389:
11387:
11386:
11381:
11379:
11377:
11372:
11371:
11362:
11357:
11352:
11351:
11342:
11340:
11335:
11327:
11307:
11305:
11304:
11299:
11291:
11286:
11285:
11276:
11268:
11267:
11249:
11248:
11239:
11238:
11219:
11215:
11207:
11205:
11204:
11199:
11191:
11189:
11188:
11187:
11174:
11173:
11164:
11153:
11152:
11136:
11135:
11133:
11132:
11126:
11123:
11102:
11101:
11099:
11098:
11092:
11089:
11071:
11038:
11036:
11035:
11030:
11025:
11020:
11019:
11011:
10999:
10979:
10977:
10976:
10971:
10966:
10964:
10956:
10948:
10933:
10931:
10930:
10925:
10920:
10912:
10879:
10877:
10876:
10871:
10860:
10859:
10844:
10843:
10824:
10822:
10821:
10816:
10805:
10804:
10789:
10788:
10766:
10764:
10763:
10758:
10756:
10754:
10749:
10748:
10746:
10741:
10740:
10731:
10726:
10721:
10720:
10711:
10709:
10704:
10696:
10690:
10685:
10683:
10678:
10677:
10675:
10670:
10669:
10660:
10655:
10650:
10649:
10640:
10638:
10633:
10625:
10619:
10604:
10602:
10601:
10596:
10569:
10567:
10566:
10561:
10556:
10554:
10549:
10548:
10546:
10541:
10540:
10531:
10526:
10521:
10520:
10511:
10509:
10504:
10496:
10490:
10476:
10474:
10473:
10468:
10463:
10461:
10456:
10455:
10453:
10448:
10447:
10438:
10433:
10428:
10427:
10418:
10416:
10411:
10403:
10397:
10382:
10380:
10379:
10374:
10362:
10358:
10356:
10355:
10350:
10345:
10343:
10338:
10337:
10328:
10323:
10318:
10317:
10308:
10306:
10301:
10293:
10281:
10279:
10278:
10273:
10271:
10266:
10265:
10264:
10251:
10246:
10245:
10223:
10221:
10220:
10215:
10207:
10202:
10201:
10192:
10178:
10177:
10161:
10159:
10158:
10153:
10151:
10147:
10146:
10119:
10118:
10106:
10105:
10090:
10089:
10074:
10070:
10069:
10060:
10059:
10041:
10040:
10028:
10027:
10012:
10011:
9996:
9989:
9988:
9967:
9966:
9924:
9922:
9921:
9916:
9911:
9910:
9894:
9886:
9884:
9883:
9878:
9876:
9869:
9861:
9826:
9825:
9813:
9812:
9792:
9790:
9789:
9784:
9752:
9750:
9749:
9744:
9691:
9690:
9678:
9677:
9661:
9659:
9658:
9653:
9629:
9627:
9626:
9621:
9609:
9605:
9580:Gerolamo Cardano
9566:Cardano's method
9549:
9545:
9526:
9517:
9507:rational numbers
9504:
9495:
9491:
9482:
9473:
9471:
9470:
9465:
9463:
9458:
9449:
9447:
9446:
9441:
9439:
9434:
9425:
9416:
9407:
9398:
9394:
9392:
9391:
9386:
9378:
9377:
9365:
9364:
9349:
9348:
9336:
9335:
9320:
9319:
9307:
9306:
9294:
9293:
9281:
9276:
9267:
9260:
9251:
9238:
9234:
9226:
9214:
9198:
9197:
9195:
9194:
9191:
9188:
9187:
9178:
9177:
9175:
9174:
9171:
9168:
9167:
9158:
9157:
9155:
9154:
9151:
9148:
9147:
9129:Marden's theorem
9113:
9106:
9098:
9091:
9081:
9061:
9060:
9053:
9052:
9045:
9044:
9043:
9033:
9032:
9031:
9021:
9017:
9016:
9005:
8978:
8974:
8970:
8966:
8962:
8961:
8959:
8958:
8952:
8949:
8931:
8930:
8928:
8927:
8921:
8918:
8900:
8883:
8881:
8880:
8875:
8873:
8871:
8863:
8858:
8846:
8839:
8838:
8836:
8835:
8832:
8829:
8828:
8814:
8813:
8811:
8810:
8807:
8804:
8796:
8794:
8793:
8788:
8786:
8782:
8781:
8775:
8767:
8766:
8764:
8762:
8754:
8746:
8724:
8722:
8721:
8716:
8688:
8685:
8682:
8678:
8677:
8672:
8664:
8656:
8652:
8651:
8645:
8637:
8636:
8634:
8632:
8624:
8616:
8603:
8595:
8582:
8580:
8572:
8567:
8559:
8558:
8540:
8510:inflection point
8507:
8499:
8495:
8491:
8487:
8466:Three real roots
8447:angle trisection
8421:
8415:
8413:
8411:
8410:
8405:
8402:
8380:
8374:
8372:
8371:
8366:
8361:
8357:
8350:
8348:
8347:
8335:
8322:
8320:
8319:
8310:
8309:
8300:
8291:
8278:
8274:
8270:
8267:on the positive
8262:
8244:
8237:
8216:
8212:
8208:
8201:
8177:
8170:
8156:
8138:
8127:
8118:
8108:
8101:
8099:
8098:
8093:
8091:
8076:
8075:
8072:
8069:
8065:
8064:
8060:
8059:
8050:
8049:
8047:
8045:
8037:
8029:
8016:
8008:
7995:
7986:
7985:
7970:
7969:
7942:
7941:
7938:
7934:
7927:
7926:
7911:
7910:
7896:
7895:
7892:
7889:
7885:
7884:
7880:
7879:
7873:
7865:
7864:
7862:
7860:
7852:
7851:
7843:
7831:
7818:
7810:
7797:
7795:
7787:
7782:
7780:
7775:
7774:
7766:
7760:
7745:
7744:
7720:
7706:
7704:
7703:
7698:
7675:
7672:
7669:
7665:
7664:
7659:
7648:
7643:
7639:
7638:
7632:
7624:
7623:
7621:
7619:
7611:
7603:
7590:
7582:
7568:
7566:
7558:
7553:
7544:
7543:
7527:
7525:
7524:
7519:
7513:
7507:
7499:
7498:
7496:
7494:
7486:
7478:
7448:
7446:
7445:
7440:
7432:
7426:
7418:
7417:
7414:
7412:
7404:
7396:
7370:
7369:
7350:
7348:
7347:
7342:
7337:
7332:
7331:
7322:
7313:
7311:
7310:
7305:
7299:
7297:
7289:
7284:
7265:
7263:
7262:
7257:
7207:
7206:
7187:
7183:
7181:
7165:
7140:
7133:
7131:
7130:
7125:
7117:
7116:
7101:
7100:
7075:
7073:
7072:
7067:
7044:
7041:
7038:
7034:
7032:
7027:
7016:
7011:
7007:
7005:
6999:
6991:
6990:
6988:
6986:
6978:
6970:
6957:
6949:
6934:
6932:
6924:
6919:
6910:
6909:
6893:
6891:
6890:
6885:
6862:
6861:
6808:
6806:
6805:
6800:
6792:
6791:
6779:
6778:
6762:
6760:
6759:
6754:
6748:
6747:
6738:
6736:
6725:
6703:
6701:
6700:
6695:
6690:
6688:
6680:
6672:
6663:
6661:
6660:
6655:
6644:
6643:
6631:
6630:
6611:
6610:
6608:
6607:
6604:
6601:
6600:
6599:
6583:
6581:
6580:
6575:
6573:
6570:
6537:
6533:
6532:
6530:
6526:
6525:
6515:
6514:
6505:
6497:
6496:
6476:
6474:
6463:
6455:
6454:
6438:
6437:
6435:
6434:
6431:
6428:
6427:
6426:
6410:
6406:
6400:
6398:
6397:
6392:
6390:
6387:
6385:
6381:
6380:
6375:
6374:
6365:
6346:
6344:
6333:
6315:
6313:
6312:
6309:
6306:
6299:
6297:
6296:
6291:
6280:
6279:
6267:
6266:
6250:
6243:
6241:
6240:
6235:
6227:
6226:
6211:(this occurs if
6210:
6203:
6199:
6195:
6188:square root and
6183:
6181:
6180:
6175:
6173:
6171:
6166:
6165:
6162:
6160:
6157:
6154:
6145:
6143:
6142:
6137:
6135:
6133:
6132:
6129:
6127:
6124:
6121:
6112:
6110:
6109:
6104:
6099:
6097:
6088:
6087:
6084:
6079:
6063:
6058:
6049:
6044:
6043:
6033:
6032:
6014:
6013:
6011:
6010:
6004:
6001:
5993:
5991:
5990:
5985:
5983:
5982:
5966:
5965:
5963:
5962:
5956:
5953:
5945:
5943:
5942:
5937:
5935:
5934:
5914:
5912:
5911:
5906:
5904:
5903:
5887:
5885:
5884:
5879:
5877:
5876:
5858:
5856:
5855:
5850:
5848:
5838:
5837:
5807:
5806:
5787:
5786:
5758:
5757:
5741:
5740:
5706:
5684:
5682:
5681:
5676:
5653:
5652:
5637:
5636:
5617:
5590:
5586:
5580:
5578:
5577:
5572:
5564:
5563:
5548:
5547:
5524:
5522:
5521:
5516:
5498:
5496:
5495:
5490:
5469:
5462:
5455:
5442:
5440:
5439:
5434:
5428:
5423:
5422:
5414:
5402:
5391:
5389:
5388:
5383:
5381:
5379:
5374:
5373:
5370:
5368:
5365:
5362:
5353:
5351:
5350:
5345:
5343:
5341:
5340:
5337:
5335:
5332:
5329:
5320:
5318:
5317:
5312:
5307:
5305:
5300:
5299:
5297:
5292:
5291:
5282:
5277:
5272:
5271:
5262:
5260:
5255:
5247:
5241:
5232:
5229:
5226:
5224:
5213:
5198:
5181:principal values
5178:
5176:
5175:
5170:
5168:
5166:
5161:
5160:
5157:
5155:
5152:
5149:
5140:
5138:
5137:
5132:
5130:
5128:
5127:
5124:
5122:
5119:
5116:
5099:irreducible case
5082:
5080:
5079:
5074:
5063:
5062:
5047:
5046:
5025:
5023:
5022:
5017:
5012:
5010:
5005:
5004:
4995:
4993:
4992:
4980:
4978:
4973:
4972:
4963:
4961:
4960:
4944:
4942:
4941:
4936:
4934:
4932:
4927:
4926:
4917:
4915:
4914:
4902:
4900:
4895:
4894:
4885:
4883:
4882:
4866:
4864:
4863:
4858:
4853:
4848:
4847:
4842:
4827:
4821:
4816:
4804:
4803:
4787:
4785:
4784:
4779:
4774:
4769:
4768:
4763:
4748:
4743:
4742:
4702:
4700:
4699:
4694:
4689:
4687:
4682:
4681:
4672:
4667:
4662:
4661:
4652:
4650:
4645:
4637:
4625:
4623:
4622:
4617:
4615:
4613:
4608:
4607:
4598:
4593:
4588:
4587:
4578:
4576:
4571:
4563:
4551:
4549:
4548:
4543:
4541:
4540:
4524:
4522:
4521:
4516:
4514:
4513:
4497:
4495:
4494:
4489:
4484:
4482:
4477:
4476:
4467:
4462:
4460:
4455:
4454:
4445:
4432:
4430:
4429:
4424:
4422:
4417:
4416:
4407:
4402:
4397:
4396:
4387:
4374:
4370:
4366:
4364:
4363:
4358:
4335:
4334:
4302:Gerolamo Cardano
4287:Vieta's formulas
4260:
4258:
4257:
4252:
4247:
4246:
4237:
4236:
4215:
4214:
4175:
4174:
4159:
4158:
4139:
4137:
4136:
4131:
4126:
4124:
4108:
4107:
4091:
4090:
4089:
4074:
4073:
4045:
4040:
4039:
4023:
4021:
4020:
4015:
4010:
4008:
3992:
3991:
3975:
3955:
3950:
3949:
3937:
3936:
3920:
3918:
3917:
3912:
3895:
3894:
3876:
3874:
3873:
3868:
3866:
3865:
3860:
3856:
3855:
3853:
3842:
3807:
3806:
3791:
3790:
3771:
3769:
3768:
3763:
3758:
3756:
3745:
3737:
3736:
3724:
3723:
3711:
3710:
3694:
3692:
3691:
3686:
3669:
3668:
3648:
3646:
3645:
3640:
3623:
3622:
3607:
3606:
3579:Vieta's formulas
3570:
3568:
3567:
3562:
3557:
3556:
3551:
3547:
3546:
3544:
3536:
3528:
3514:
3510:
3509:
3504:
3496:
3465:
3464:
3449:In other words,
3446:
3444:
3443:
3438:
3433:
3431:
3423:
3415:
3407:
3406:
3394:
3393:
3375:
3373:
3372:
3367:
3365:
3360:
3352:
3347:
3346:
3330:
3323:
3321:
3320:
3315:
3304:
3303:
3288:
3287:
3268:
3257:
3253:
3251:
3250:
3245:
3237:
3236:
3221:
3220:
3201:
3199:
3198:
3193:
3176:
3175:
3145:
3143:
3142:
3137:
3119:
3103:
3087:
3070:, and thus that
3065:
3049:
3040:
3032:, is real, then
3031:
3020:
3018:
3017:
3012:
2992:
2990:
2989:
2984:
2982:
2981:
2972:
2971:
2959:
2958:
2946:
2945:
2936:
2935:
2923:
2922:
2910:
2909:
2900:
2899:
2887:
2886:
2874:
2873:
2851:
2849:
2848:
2843:
2826:
2825:
2810:
2809:
2790:
2756:
2744:
2742:
2741:
2736:
2709:
2707:
2706:
2701:
2676:
2674:
2673:
2668:
2644:
2640:
2616:Vieta's formulas
2604:
2602:
2601:
2596:
2591:
2589:
2588:
2587:
2574:
2573:
2572:
2560:
2559:
2529:
2528:
2510:
2509:
2488:
2487:
2471:
2461:
2459:
2458:
2453:
2451:
2450:
2434:
2432:
2431:
2426:
2421:
2420:
2411:
2410:
2394:
2393:
2374:
2373:
2364:
2363:
2348:
2347:
2308:
2306:
2305:
2300:
2283:
2282:
2267:
2266:
2245:
2243:
2242:
2237:
2232:
2228:
2227:
2226:
2210:
2209:
2181:
2179:
2178:
2173:
2156:
2155:
2137:
2135:
2134:
2129:
2124:
2123:
2114:
2113:
2101:
2100:
2088:
2087:
2078:
2077:
2065:
2064:
2052:
2051:
2042:
2041:
2029:
2028:
2016:
2015:
1999:
1997:
1996:
1991:
1971:
1970:
1955:
1954:
1927:
1861:
1859:
1858:
1853:
1823:
1821:
1820:
1815:
1810:
1808:
1797:
1792:
1791:
1779:
1778:
1762:
1760:
1759:
1754:
1752:
1751:
1739:
1738:
1726:
1725:
1709:
1707:
1706:
1701:
1699:
1698:
1686:
1685:
1673:
1672:
1651:
1649:
1648:
1643:
1641:
1634:
1632:
1631:
1630:
1617:
1613:
1612:
1582:
1581:
1568:
1564:
1552:
1550:
1549:
1548:
1535:
1534:
1533:
1511:
1507:
1495:
1493:
1482:
1472:
1453:
1451:
1450:
1445:
1419:
1418:
1399:
1392:
1386:
1382:
1380:
1379:
1374:
1372:
1370:
1359:
1338:
1336:
1335:
1330:
1307:
1306:
1291:
1290:
1264:
1262:
1261:
1256:
1239:
1238:
1203:
1193:
1191:
1190:
1185:
1183:
1181:
1180:
1171:
1170:
1169:
1142:
1141:
1128:
1120:
1118:
1117:
1108:
1091:
1086:
1085:
1076:
1068:
1055:
1053:
1052:
1047:
1041:
1033:
1028:
1027:
1007:
1003:
999:
995:
991:
987:
978:coprime integers
975:
971:
967:
965:
964:
959:
923:
921:
920:
915:
889:
888:
873:
872:
848:rational numbers
811:Lodovico Ferrari
797:Gerolamo Cardano
791:
777:
742:negative numbers
739:
735:
731:
702:
638:
598:
591:
580:
501:are 0, 1 or 2.
477:). Hippocrates,
345:
341:
337:
333:
315:
311:
309:
308:
303:
280:
279:
264:
263:
228:
226:
225:
220:
218:
169:
161:
153:
149:
145:
129:
128:
113:
112:
98:
90:
54:
16179:
16178:
16174:
16173:
16172:
16170:
16169:
16168:
16144:
16143:
16142:
16137:
16086:
16025:
15968:Linear equation
15938:
15929:
15855:
15852:
15808:
15768:
15763:
15746:
15706:
15666:
15661:
15645:
15616:10.2307/3620218
15597:
15568:10.2307/3619617
15549:
15543:
15526:
15523:
15521:Further reading
15506:10.2307/3027812
15488:
15483:
15459:
15452:
15442:
15426:
15422:
15412:
15396:
15392:
15374:
15370:
15364:
15354:Springer-Verlag
15343:
15339:
15325:10.2307/1967772
15309:
15305:
15291:10.2307/2972885
15273:
15269:
15255:10.2307/2301359
15237:
15233:
15202:10.2307/3619777
15185:
15179:
15175:
15167:
15153:10.2307/2323624
15136:
15130:
15126:
15117:
15113:
15102:
15098:
15089:
15085:
15078:
15062:
15058:
15025:
15021:
15012:
15010:
15002:
15001:
14997:
14987:
14971:
14967:
14933:
14927:
14918:
14882:
14878:
14871:
14851:
14847:
14823:
14819:
14789:
14785:
14761:
14757:
14751:
14732:
14728:
14721:
14705:
14701:
14693:
14689:
14676:
14672:
14667:
14663:
14657:
14640:
14636:
14629:
14612:
14608:
14601:
14578:
14574:
14570:, pp. 8–9)
14566:
14559:
14551:
14547:
14540:
14524:
14520:
14513:
14491:
14487:
14480:
14464:
14460:
14448:
14441:
14434:
14418:
14411:
14405:
14379:
14375:
14371:
14366:
14365:
14354:
14339:
14336:
14331:
14330:
14328:
14323:
14315:
14312:
14307:
14306:
14304:
14291:
14287:
14285:
14281:
14252:
14226:
14222:
14217:
14201:cubic equation.
14154:
14131:Critical points
14084:resolvent cubic
14050:
14042:
14039:
14038:
13973:
13965:
13950:
13947:
13942:
13941:
13939:
13937:
13931:
13921:
13918:
13913:
13912:
13910:
13908:
13902:
13892:
13889:
13884:
13883:
13881:
13879:
13873:
13866:
13860:
13853:
13847:
13843:
13837:
13823:
13822:
13813:
13809:
13797:
13793:
13787:
13783:
13771:
13766:
13752:
13746:
13745:
13736:
13732:
13720:
13715:
13704:
13697:
13695:
13692:
13691:
13679:
13675:
13661:
13660:
13651:
13647:
13641:
13637:
13631:
13627:
13612:
13607:
13597:
13593:
13584:
13579:
13569:
13565:
13556:
13551:
13541:
13537:
13528:
13524:
13518:
13513:
13500:
13496:
13490:
13485:
13472:
13468:
13462:
13457:
13435:
13430:
13417:
13412:
13399:
13394:
13375:
13370:
13357:
13352:
13341:
13335:
13334:
13322:
13318:
13312:
13308:
13299:
13295:
13289:
13285:
13276:
13272:
13266:
13262:
13250:
13245:
13232:
13227:
13214:
13209:
13196:
13192:
13186:
13182:
13175:
13168:
13166:
13163:
13162:
13151:
13144:
13141:
13137:
13133:
13132:Computation of
13099:
13095:
13093:
13090:
13089:
13063:
13059:
13057:
13054:
13053:
13034:
13031:
13030:
13026:
12997:
12983:
12980:
12979:
12950:
12946:
12944:
12941:
12940:
12908:
12904:
12898:
12894:
12892:
12889:
12888:
12868:
12864:
12855:
12851:
12836:
12827:
12823:
12821:
12818:
12817:
12796:
12793:
12792:
12788:
12771:
12767:
12765:
12762:
12761:
12744:
12740:
12738:
12735:
12734:
12711:
12707:
12689:
12685:
12683:
12680:
12679:
12662:
12657:
12651:
12648:
12647:
12630:
12625:
12619:
12616:
12615:
12598:
12593:
12580:
12575:
12563:
12560:
12559:
12539:
12535:
12529:
12525:
12517:
12514:
12513:
12507:
12503:
12500:
12496:
12492:
12486:
12483:
12477:
12474:
12468:
12465:
12459:
12458:exchanges also
12456:
12450:
12447:
12441:
12434:
12428:
12425:
12419:
12409:
12406:
12401:
12400:
12398:
12396:
12393:
12387:
12367:
12366:
12354:
12350:
12344:
12340:
12331:
12327:
12315:
12311:
12296:
12289:
12283:
12279:
12276:
12275:
12263:
12259:
12247:
12243:
12237:
12233:
12224:
12220:
12205:
12198:
12192:
12188:
12185:
12184:
12172:
12168:
12159:
12155:
12146:
12142:
12127:
12120:
12114:
12110:
12106:
12104:
12101:
12100:
12097:
12091:
12088:
12082:
12079:
12073:
12059:
12058:
12049:
12045:
12033:
12029:
12023:
12019:
12010:
12006:
11999:
11993:
11989:
11986:
11985:
11976:
11972:
11966:
11962:
11953:
11949:
11937:
11933:
11926:
11920:
11916:
11913:
11912:
11903:
11899:
11890:
11886:
11877:
11873:
11866:
11860:
11856:
11852:
11850:
11847:
11846:
11843:
11837:
11834:
11828:
11825:
11819:
11794:
11784:
11780:
11764:
11751:
11749:
11740:
11737:
11736:
11733:complex numbers
11721:
11714:
11706:
11684:
11665:
11639:
11630:
11619:
11612:
11605:
11596:
11593:
11588:
11587:
11585:
11583:
11580:
11572:
11547:
11541:
11537:
11535:
11529:
11526:
11525:
11518:
11511:
11508:
11503:
11502:
11500:
11498:
11492:
11485:
11478:
11475:
11470:
11469:
11467:
11465:
11459:
11452:
11445:
11442:
11437:
11436:
11434:
11432:
11426:
11422:
11415:
11409:
11406:
11400:
11397:
11391:
11367:
11363:
11361:
11347:
11343:
11341:
11339:
11326:
11315:
11312:
11311:
11281:
11277:
11275:
11263:
11259:
11244:
11240:
11234:
11230:
11225:
11222:
11221:
11217:
11211:
11210:Multiplying by
11183:
11179:
11175:
11169:
11165:
11163:
11148:
11144:
11142:
11139:
11138:
11127:
11124:
11119:
11118:
11116:
11107:
11093:
11090:
11085:
11084:
11082:
11073:
11058:
11044:
11010:
11000:
10998:
10996:
10993:
10992:
10957:
10949:
10947:
10939:
10936:
10935:
10911:
10897:
10894:
10893:
10886:Rafael Bombelli
10855:
10851:
10839:
10835:
10830:
10827:
10826:
10800:
10796:
10784:
10780:
10775:
10772:
10771:
10750:
10736:
10732:
10730:
10716:
10712:
10710:
10708:
10695:
10691:
10689:
10679:
10665:
10661:
10659:
10645:
10641:
10639:
10637:
10624:
10620:
10618:
10610:
10607:
10606:
10575:
10572:
10571:
10550:
10536:
10532:
10530:
10516:
10512:
10510:
10508:
10495:
10491:
10489:
10481:
10478:
10477:
10457:
10443:
10439:
10437:
10423:
10419:
10417:
10415:
10402:
10398:
10396:
10388:
10385:
10384:
10368:
10365:
10364:
10360:
10333:
10329:
10327:
10313:
10309:
10307:
10305:
10292:
10287:
10284:
10283:
10260:
10256:
10252:
10250:
10241:
10237:
10229:
10226:
10225:
10197:
10193:
10191:
10173:
10169:
10167:
10164:
10163:
10149:
10148:
10142:
10138:
10114:
10110:
10101:
10097:
10085:
10081:
10072:
10071:
10065:
10061:
10055:
10051:
10036:
10032:
10023:
10019:
10007:
10003:
9994:
9993:
9984:
9980:
9962:
9958:
9942:
9935:
9933:
9930:
9929:
9906:
9902:
9900:
9897:
9896:
9890:
9874:
9873:
9860:
9850:
9841:
9840:
9827:
9821:
9817:
9808:
9804:
9800:
9798:
9795:
9794:
9760:
9757:
9756:
9686:
9682:
9673:
9669:
9667:
9664:
9663:
9635:
9632:
9631:
9615:
9612:
9611:
9607:
9592:
9568:
9556:
9547:
9528:
9525:
9519:
9516:
9510:
9500:
9493:
9490:
9484:
9481:
9475:
9457:
9455:
9452:
9451:
9433:
9431:
9428:
9427:
9424:
9418:
9415:
9409:
9406:
9400:
9396:
9373:
9369:
9360:
9356:
9344:
9340:
9331:
9327:
9315:
9311:
9302:
9298:
9289:
9285:
9275:
9273:
9270:
9269:
9265:
9259:
9253:
9250:
9244:
9241:splitting field
9236:
9232:
9224:
9212:
9205:
9192:
9189:
9185:
9184:
9183:
9181:
9180:
9172:
9169:
9165:
9164:
9163:
9161:
9160:
9152:
9149:
9145:
9144:
9143:
9141:
9140:
9120:
9111:
9109:imaginary parts
9104:
9096:
9083:
9079:
9072:Cartesian plane
9056:
9055:
9048:
9047:
9038:
9036:
9035:
9026:
9024:
9023:
9019:
9012:
9007:
8997:
8990:
8985:
8976:
8972:
8968:
8964:
8953:
8950:
8945:
8944:
8942:
8933:
8922:
8919:
8914:
8913:
8911:
8902:
8898:
8862:
8857:
8852:
8849:
8848:
8841:
8833:
8830:
8826:
8824:
8823:
8821:
8816:
8808:
8805:
8802:
8801:
8799:
8798:
8768:
8765:
8755:
8747:
8745:
8744:
8740:
8732:
8729:
8728:
8684:
8665:
8663:
8638:
8635:
8625:
8617:
8615:
8614:
8610:
8594:
8593:
8589:
8571:
8566:
8554:
8550:
8548:
8545:
8544:
8527:
8505:
8497:
8493:
8489:
8485:
8468:
8463:
8436:angle trisector
8428:
8422:on the circle.
8417:
8406:
8403:
8398:
8397:
8395:
8382:
8376:
8343:
8339:
8334:
8333:
8329:
8315:
8311:
8305:
8301:
8299:
8297:
8294:
8293:
8283:
8276:
8272:
8268:
8250:
8239:
8222:
8214:
8210:
8203:
8196:
8189:
8184:
8172:
8164:
8158:
8150:
8144:
8133:
8126:
8120:
8113:
8103:
8089:
8088:
8071:
8048:
8038:
8030:
8028:
8027:
8023:
8007:
8006:
8002:
7984:
7971:
7965:
7961:
7958:
7957:
7939: and
7937:
7922:
7918:
7906:
7902:
7891:
7866:
7863:
7853:
7847:
7839:
7832:
7830:
7829:
7825:
7809:
7808:
7804:
7786:
7781:
7770:
7762:
7761:
7759:
7746:
7740:
7736:
7732:
7730:
7727:
7726:
7715:
7712:
7671:
7649:
7647:
7625:
7622:
7612:
7604:
7602:
7601:
7597:
7581:
7580:
7576:
7557:
7552:
7539:
7535:
7533:
7530:
7529:
7500:
7497:
7487:
7479:
7477:
7454:
7451:
7450:
7419:
7416:
7405:
7397:
7395:
7365:
7361:
7356:
7353:
7352:
7327:
7323:
7321:
7319:
7316:
7315:
7288:
7283:
7271:
7268:
7267:
7202:
7198:
7193:
7190:
7189:
7185:
7169:
7167:
7152:
7135:
7112:
7108:
7096:
7092:
7087:
7084:
7083:
7040:
7017:
7015:
6992:
6989:
6979:
6971:
6969:
6968:
6964:
6948:
6946:
6942:
6923:
6918:
6905:
6901:
6899:
6896:
6895:
6857:
6853:
6851:
6848:
6847:
6845:depressed cubic
6819:
6814:
6787:
6783:
6774:
6770:
6768:
6765:
6764:
6743:
6739:
6729:
6724:
6709:
6706:
6705:
6681:
6673:
6671:
6669:
6666:
6665:
6639:
6635:
6626:
6622:
6620:
6617:
6616:
6605:
6602:
6597:
6595:
6593:
6592:
6590:
6585:
6569:
6521:
6517:
6516:
6510:
6506:
6504:
6492:
6488:
6481:
6477:
6467:
6462:
6450:
6446:
6444:
6441:
6440:
6432:
6429:
6424:
6422:
6420:
6419:
6417:
6416:
6408:
6404:
6386:
6370:
6366:
6364:
6351:
6347:
6337:
6332:
6321:
6318:
6317:
6310:
6307:
6304:
6303:
6301:
6275:
6271:
6262:
6258:
6256:
6253:
6252:
6245:
6222:
6218:
6216:
6213:
6212:
6205:
6201:
6197:
6193:
6167:
6161:
6156:
6155:
6153:
6151:
6148:
6147:
6128:
6123:
6122:
6120:
6118:
6115:
6114:
6093:
6080:
6075:
6059:
6054:
6048:
6039:
6035:
6034:
6031:
6023:
6020:
6019:
6005:
6002:
5999:
5998:
5996:
5995:
5978:
5974:
5972:
5969:
5968:
5957:
5954:
5951:
5950:
5948:
5947:
5930:
5926:
5924:
5921:
5920:
5899:
5895:
5893:
5890:
5889:
5872:
5868:
5866:
5863:
5862:
5846:
5845:
5833:
5829:
5802:
5798:
5788:
5782:
5778:
5775:
5774:
5753:
5749:
5742:
5736:
5732:
5728:
5726:
5723:
5722:
5690:
5687:depressed cubic
5648:
5644:
5632:
5628:
5623:
5620:
5619:
5612:
5605:
5588:
5584:
5559:
5555:
5543:
5539:
5534:
5531:
5530:
5504:
5501:
5500:
5475:
5472:
5471:
5464:
5457:
5446:
5413:
5403:
5401:
5398:
5395:
5394:
5375:
5369:
5364:
5363:
5361:
5359:
5356:
5355:
5336:
5331:
5330:
5328:
5326:
5323:
5322:
5301:
5287:
5283:
5281:
5267:
5263:
5261:
5259:
5246:
5242:
5240:
5228:
5217:
5212:
5204:
5201:
5200:
5192:
5162:
5156:
5151:
5150:
5148:
5146:
5143:
5142:
5123:
5118:
5117:
5115:
5113:
5110:
5109:
5058:
5054:
5042:
5038:
5033:
5030:
5029:
5006:
5000:
4996:
4994:
4988:
4984:
4974:
4968:
4964:
4962:
4956:
4952:
4950:
4947:
4946:
4928:
4922:
4918:
4916:
4910:
4906:
4896:
4890:
4886:
4884:
4878:
4874:
4872:
4869:
4868:
4841:
4828:
4826:
4817:
4812:
4799:
4795:
4793:
4790:
4789:
4762:
4749:
4747:
4738:
4734:
4732:
4729:
4728:
4722:complex numbers
4677:
4673:
4671:
4657:
4653:
4651:
4649:
4636:
4631:
4628:
4627:
4603:
4599:
4597:
4583:
4579:
4577:
4575:
4562:
4557:
4554:
4553:
4536:
4532:
4530:
4527:
4526:
4509:
4505:
4503:
4500:
4499:
4478:
4472:
4468:
4466:
4456:
4450:
4446:
4444:
4442:
4439:
4438:
4412:
4408:
4406:
4392:
4388:
4386:
4384:
4381:
4380:
4372:
4368:
4330:
4326:
4324:
4321:
4320:
4299:
4267:
4242:
4238:
4232:
4228:
4210:
4206:
4170:
4166:
4154:
4150:
4145:
4142:
4141:
4103:
4099:
4092:
4085:
4081:
4069:
4065:
4046:
4044:
4035:
4031:
4029:
4026:
4025:
3987:
3983:
3976:
3956:
3954:
3945:
3941:
3932:
3928:
3926:
3923:
3922:
3890:
3886:
3884:
3881:
3880:
3861:
3846:
3841:
3834:
3830:
3829:
3802:
3798:
3786:
3782:
3777:
3774:
3773:
3749:
3744:
3732:
3728:
3719:
3715:
3706:
3702:
3700:
3697:
3696:
3664:
3660:
3658:
3655:
3654:
3618:
3614:
3602:
3598:
3593:
3590:
3589:
3577:resulting from
3552:
3537:
3529:
3527:
3520:
3516:
3515:
3497:
3495:
3488:
3484:
3460:
3456:
3454:
3451:
3450:
3424:
3416:
3414:
3402:
3398:
3389:
3385:
3383:
3380:
3379:
3353:
3351:
3342:
3338:
3336:
3333:
3332:
3325:
3299:
3295:
3283:
3279:
3274:
3271:
3270:
3259:
3255:
3232:
3228:
3216:
3212:
3207:
3204:
3203:
3171:
3167:
3165:
3162:
3161:
3151:
3125:
3122:
3121:
3118:
3111:
3105:
3102:
3095:
3089:
3085:
3078:
3071:
3064:
3057:
3051:
3048:
3042:
3039:
3033:
3030:
3024:
3000:
2997:
2996:
2977:
2973:
2967:
2963:
2954:
2950:
2941:
2937:
2931:
2927:
2918:
2914:
2905:
2901:
2895:
2891:
2882:
2878:
2869:
2865:
2857:
2854:
2853:
2821:
2817:
2805:
2801:
2796:
2793:
2792:
2789:
2782:
2775:
2769:
2754:
2721:
2718:
2717:
2686:
2683:
2682:
2662:
2659:
2658:
2651:
2642:
2639:
2632:
2625:
2619:
2583:
2579:
2575:
2568:
2564:
2555:
2551:
2524:
2520:
2505:
2501:
2483:
2479:
2472:
2470:
2468:
2465:
2464:
2446:
2442:
2440:
2437:
2436:
2416:
2412:
2406:
2402:
2389:
2385:
2369:
2365:
2359:
2355:
2343:
2339:
2314:
2311:
2310:
2278:
2274:
2262:
2258:
2253:
2250:
2249:
2222:
2218:
2205:
2201:
2196:
2192:
2187:
2184:
2183:
2151:
2147:
2145:
2142:
2141:
2119:
2115:
2109:
2105:
2096:
2092:
2083:
2079:
2073:
2069:
2060:
2056:
2047:
2043:
2037:
2033:
2024:
2020:
2011:
2007:
2005:
2002:
2001:
1966:
1962:
1950:
1946:
1941:
1938:
1937:
1936:) of the cubic
1926:
1919:
1912:
1906:
1884:
1868:
1829:
1826:
1825:
1801:
1796:
1787:
1783:
1774:
1770:
1768:
1765:
1764:
1747:
1743:
1734:
1730:
1721:
1717:
1715:
1712:
1711:
1694:
1690:
1681:
1677:
1668:
1664:
1662:
1659:
1658:
1639:
1638:
1626:
1622:
1618:
1608:
1604:
1577:
1573:
1569:
1567:
1565:
1563:
1554:
1553:
1544:
1540:
1536:
1529:
1525:
1512:
1510:
1508:
1506:
1497:
1496:
1486:
1481:
1473:
1471:
1461:
1459:
1456:
1455:
1414:
1410:
1408:
1405:
1404:
1397:
1388:
1384:
1363:
1358:
1344:
1341:
1340:
1302:
1298:
1286:
1282:
1277:
1274:
1273:
1234:
1230:
1228:
1225:
1224:
1221:
1219:Depressed cubic
1195:
1176:
1172:
1165:
1161:
1137:
1133:
1129:
1127:
1113:
1109:
1092:
1090:
1081:
1077:
1067:
1065:
1062:
1061:
1032:
1023:
1019:
1016:
1013:
1012:
1005:
1001:
997:
993:
989:
985:
984:allows finding
973:
969:
944:
941:
940:
884:
880:
868:
864:
859:
856:
855:
852:common multiple
844:
826:Rafael Bombelli
818:complex numbers
779:
765:
737:
733:
719:
689:
643:mathematician,
625:
593:
582:
564:
431:
423:complex numbers
406:Newton's method
343:
339:
335:
331:
313:
275:
271:
259:
255:
250:
247:
246:
216:
215:
160:
151:
150:
124:
120:
108:
104:
103:
99:
89:
82:
66:
64:
61:
60:
57:critical points
49:
35:
28:
23:variables, see
17:
12:
11:
5:
16177:
16167:
16166:
16161:
16156:
16139:
16138:
16136:
16135:
16130:
16125:
16120:
16115:
16110:
16105:
16100:
16094:
16092:
16088:
16087:
16085:
16084:
16079:
16074:
16069:
16064:
16059:
16054:
16049:
16044:
16039:
16033:
16031:
16027:
16026:
16024:
16023:
16018:
16013:
16008:
16007:
16006:
15996:
15995:
15994:
15992:Cubic equation
15984:
15983:
15982:
15972:
15971:
15970:
15960:
15955:
15949:
15947:
15940:
15939:
15928:
15927:
15920:
15913:
15905:
15899:
15898:
15881:
15871:
15851:
15850:External links
15848:
15847:
15846:
15806:
15766:
15761:
15744:
15704:
15664:
15659:
15643:
15595:
15547:
15541:
15522:
15519:
15518:
15517:
15487:
15484:
15482:
15481:
15450:
15440:
15420:
15410:
15390:
15368:
15362:
15337:
15319:(2): 152–158,
15303:
15285:(6): 268–269,
15267:
15249:(6): 383–384,
15231:
15173:
15170:on 2015-12-19.
15147:(3): 185–194.
15124:
15111:
15096:
15083:
15076:
15056:
15019:
14995:
14985:
14965:
14916:
14876:
14869:
14845:
14817:
14805:10.2307/604533
14799:(2): 304–309,
14783:
14755:
14749:
14726:
14719:
14699:
14695:Guilbeau (1930
14687:
14670:
14661:
14655:
14643:Mikami, Yoshio
14634:
14628:978-1603860512
14627:
14606:
14600:978-1578987542
14599:
14572:
14568:Guilbeau (1930
14557:
14553:Guilbeau (1930
14545:
14538:
14518:
14511:
14485:
14478:
14458:
14439:
14432:
14409:
14403:
14372:
14370:
14367:
14364:
14363:
14279:
14235:"Omar Khayyam"
14219:
14218:
14216:
14213:
14212:
14211:
14205:
14202:
14195:
14185:
14174:thermodynamics
14170:
14153:
14150:
14149:
14148:
14138:
14128:
14121:
14102:
14101:of the matrix.
14087:
14076:
14069:
14057:
14053:
14049:
14046:
14014:
13992:
13972:
13971:In mathematics
13969:
13964:
13961:
13935:
13906:
13877:
13864:
13851:
13841:
13821:
13816:
13812:
13808:
13805:
13800:
13796:
13790:
13786:
13782:
13779:
13774:
13769:
13765:
13761:
13758:
13755:
13753:
13751:
13748:
13747:
13744:
13739:
13735:
13731:
13728:
13723:
13718:
13714:
13710:
13707:
13705:
13703:
13700:
13699:
13659:
13654:
13650:
13644:
13640:
13634:
13630:
13626:
13623:
13620:
13615:
13610:
13606:
13600:
13596:
13592:
13587:
13582:
13578:
13572:
13568:
13564:
13559:
13554:
13550:
13544:
13540:
13536:
13531:
13527:
13521:
13516:
13512:
13508:
13503:
13499:
13493:
13488:
13484:
13480:
13475:
13471:
13465:
13460:
13456:
13452:
13449:
13446:
13443:
13438:
13433:
13429:
13425:
13420:
13415:
13411:
13407:
13402:
13397:
13393:
13389:
13386:
13383:
13378:
13373:
13369:
13365:
13360:
13355:
13351:
13347:
13344:
13342:
13340:
13337:
13336:
13333:
13330:
13325:
13321:
13315:
13311:
13307:
13302:
13298:
13292:
13288:
13284:
13279:
13275:
13269:
13265:
13261:
13258:
13253:
13248:
13244:
13240:
13235:
13230:
13226:
13222:
13217:
13212:
13208:
13204:
13199:
13195:
13189:
13185:
13181:
13178:
13176:
13174:
13171:
13170:
13140:
13130:
13116:
13113:
13110:
13107:
13102:
13098:
13077:
13074:
13071:
13066:
13062:
13041:
13038:
13014:
13011:
13005:
13002:
12996:
12993:
12990:
12987:
12967:
12964:
12961:
12958:
12953:
12949:
12928:
12925:
12922:
12919:
12916:
12911:
12907:
12901:
12897:
12876:
12871:
12867:
12863:
12858:
12854:
12850:
12844:
12841:
12835:
12830:
12826:
12803:
12800:
12774:
12770:
12747:
12743:
12722:
12719:
12714:
12710:
12706:
12703:
12700:
12697:
12692:
12688:
12665:
12660:
12656:
12633:
12628:
12624:
12601:
12596:
12592:
12588:
12583:
12578:
12574:
12570:
12567:
12547:
12542:
12538:
12532:
12528:
12524:
12521:
12505:
12498:
12490:
12481:
12472:
12463:
12454:
12445:
12432:
12423:
12391:
12365:
12362:
12357:
12353:
12347:
12343:
12339:
12334:
12330:
12326:
12323:
12318:
12314:
12310:
12304:
12301:
12295:
12292:
12290:
12286:
12282:
12278:
12277:
12274:
12271:
12266:
12262:
12258:
12255:
12250:
12246:
12240:
12236:
12232:
12227:
12223:
12219:
12213:
12210:
12204:
12201:
12199:
12195:
12191:
12187:
12186:
12183:
12180:
12175:
12171:
12167:
12162:
12158:
12154:
12149:
12145:
12141:
12135:
12132:
12126:
12123:
12121:
12117:
12113:
12109:
12108:
12095:
12086:
12077:
12057:
12052:
12048:
12044:
12041:
12036:
12032:
12026:
12022:
12018:
12013:
12009:
12005:
12002:
12000:
11996:
11992:
11988:
11987:
11984:
11979:
11975:
11969:
11965:
11961:
11956:
11952:
11948:
11945:
11940:
11936:
11932:
11929:
11927:
11923:
11919:
11915:
11914:
11911:
11906:
11902:
11898:
11893:
11889:
11885:
11880:
11876:
11872:
11869:
11867:
11863:
11859:
11855:
11854:
11841:
11832:
11823:
11806:
11801:
11797:
11793:
11790:
11787:
11783:
11779:
11774:
11768:
11763:
11760:
11757:
11754:
11748:
11745:
11638:
11635:
11601:
11576:
11559:
11553:
11550:
11544:
11540:
11534:
11516:
11496:
11483:
11463:
11450:
11430:
11417:are the three
11413:
11404:
11395:
11375:
11370:
11366:
11360:
11355:
11350:
11346:
11338:
11333:
11330:
11325:
11322:
11319:
11297:
11294:
11289:
11284:
11280:
11274:
11271:
11266:
11262:
11258:
11255:
11252:
11247:
11243:
11237:
11233:
11229:
11197:
11194:
11186:
11182:
11178:
11172:
11168:
11162:
11159:
11156:
11151:
11147:
11048:François Viète
11043:
11040:
11028:
11023:
11017:
11014:
11009:
11006:
11003:
10969:
10963:
10960:
10955:
10952:
10946:
10943:
10923:
10918:
10915:
10910:
10907:
10904:
10901:
10882:complex number
10869:
10866:
10863:
10858:
10854:
10850:
10847:
10842:
10838:
10834:
10814:
10811:
10808:
10803:
10799:
10795:
10792:
10787:
10783:
10779:
10753:
10744:
10739:
10735:
10729:
10724:
10719:
10715:
10707:
10702:
10699:
10694:
10688:
10682:
10673:
10668:
10664:
10658:
10653:
10648:
10644:
10636:
10631:
10628:
10623:
10617:
10614:
10594:
10591:
10588:
10585:
10582:
10579:
10559:
10553:
10544:
10539:
10535:
10529:
10524:
10519:
10515:
10507:
10502:
10499:
10494:
10488:
10485:
10466:
10460:
10451:
10446:
10442:
10436:
10431:
10426:
10422:
10414:
10409:
10406:
10401:
10395:
10392:
10372:
10348:
10341:
10336:
10332:
10326:
10321:
10316:
10312:
10304:
10299:
10296:
10291:
10269:
10263:
10259:
10255:
10249:
10244:
10240:
10236:
10233:
10213:
10210:
10205:
10200:
10196:
10190:
10187:
10184:
10181:
10176:
10172:
10145:
10141:
10137:
10134:
10131:
10128:
10125:
10122:
10117:
10113:
10109:
10104:
10100:
10096:
10093:
10088:
10084:
10080:
10077:
10075:
10073:
10068:
10064:
10058:
10054:
10050:
10047:
10044:
10039:
10035:
10031:
10026:
10022:
10018:
10015:
10010:
10006:
10002:
9999:
9997:
9995:
9992:
9987:
9983:
9979:
9976:
9973:
9970:
9965:
9961:
9957:
9954:
9951:
9948:
9945:
9943:
9941:
9938:
9937:
9914:
9909:
9905:
9872:
9867:
9864:
9859:
9856:
9853:
9851:
9849:
9846:
9843:
9842:
9839:
9836:
9833:
9830:
9828:
9824:
9820:
9816:
9811:
9807:
9803:
9802:
9782:
9779:
9776:
9773:
9770:
9767:
9764:
9742:
9739:
9736:
9733:
9730:
9727:
9724:
9721:
9718:
9715:
9712:
9709:
9706:
9703:
9700:
9697:
9694:
9689:
9685:
9681:
9676:
9672:
9651:
9648:
9645:
9642:
9639:
9619:
9567:
9564:
9555:
9552:
9523:
9514:
9488:
9479:
9461:
9437:
9422:
9413:
9404:
9384:
9381:
9376:
9372:
9368:
9363:
9359:
9355:
9352:
9347:
9343:
9339:
9334:
9330:
9326:
9323:
9318:
9314:
9310:
9305:
9301:
9297:
9292:
9288:
9284:
9279:
9257:
9248:
9217:characteristic
9207:Given a cubic
9204:
9201:
9119:
9116:
9068:cubic function
9039:slope of line
8989:
8986:
8984:
8981:
8869:
8866:
8861:
8856:
8785:
8778:
8774:
8771:
8761:
8758:
8753:
8750:
8743:
8739:
8736:
8714:
8710:
8707:
8704:
8701:
8698:
8695:
8692:
8681:
8675:
8671:
8668:
8662:
8659:
8655:
8648:
8644:
8641:
8631:
8628:
8623:
8620:
8613:
8609:
8606:
8601:
8598:
8592:
8588:
8585:
8578:
8575:
8570:
8565:
8562:
8557:
8553:
8474:For the cubic
8467:
8464:
8462:
8459:
8427:
8424:
8364:
8360:
8356:
8353:
8346:
8342:
8338:
8332:
8328:
8325:
8318:
8314:
8308:
8304:
8188:
8185:
8183:
8180:
8162:
8148:
8124:
8087:
8084:
8081:
8068:
8063:
8056:
8053:
8044:
8041:
8036:
8033:
8026:
8022:
8019:
8014:
8011:
8005:
8001:
7998:
7992:
7989:
7983:
7980:
7977:
7974:
7972:
7968:
7964:
7960:
7959:
7956:
7953:
7950:
7947:
7933:
7930:
7925:
7921:
7917:
7914:
7909:
7905:
7901:
7888:
7883:
7876:
7872:
7869:
7859:
7856:
7850:
7846:
7842:
7838:
7835:
7828:
7824:
7821:
7816:
7813:
7807:
7803:
7800:
7793:
7790:
7785:
7778:
7773:
7769:
7765:
7758:
7755:
7752:
7749:
7747:
7743:
7739:
7735:
7734:
7711:
7708:
7696:
7693:
7690:
7687:
7684:
7681:
7678:
7668:
7662:
7658:
7655:
7652:
7646:
7642:
7635:
7631:
7628:
7618:
7615:
7610:
7607:
7600:
7596:
7593:
7588:
7585:
7579:
7575:
7572:
7564:
7561:
7556:
7550:
7547:
7542:
7538:
7517:
7510:
7506:
7503:
7493:
7490:
7485:
7482:
7476:
7473:
7470:
7467:
7464:
7461:
7458:
7438:
7435:
7429:
7425:
7422:
7411:
7408:
7403:
7400:
7394:
7391:
7388:
7385:
7382:
7379:
7376:
7373:
7368:
7364:
7360:
7340:
7335:
7330:
7326:
7303:
7295:
7292:
7287:
7281:
7278:
7275:
7255:
7252:
7249:
7246:
7243:
7240:
7237:
7234:
7231:
7228:
7225:
7222:
7219:
7216:
7213:
7210:
7205:
7201:
7197:
7123:
7120:
7115:
7111:
7107:
7104:
7099:
7095:
7091:
7080:François Viète
7065:
7062:
7059:
7056:
7053:
7050:
7047:
7037:
7030:
7026:
7023:
7020:
7014:
7010:
7002:
6998:
6995:
6985:
6982:
6977:
6974:
6967:
6963:
6960:
6955:
6952:
6945:
6941:
6938:
6930:
6927:
6922:
6916:
6913:
6908:
6904:
6883:
6880:
6877:
6874:
6871:
6868:
6865:
6860:
6856:
6818:
6815:
6813:
6810:
6798:
6795:
6790:
6786:
6782:
6777:
6773:
6751:
6746:
6742:
6735:
6732:
6728:
6723:
6720:
6717:
6714:
6693:
6687:
6684:
6679:
6676:
6653:
6650:
6647:
6642:
6638:
6634:
6629:
6625:
6568:
6565:
6562:
6559:
6556:
6553:
6550:
6547:
6544:
6540:
6536:
6529:
6524:
6520:
6513:
6509:
6503:
6500:
6495:
6491:
6487:
6484:
6480:
6473:
6470:
6466:
6461:
6458:
6453:
6449:
6384:
6378:
6373:
6369:
6363:
6360:
6357:
6354:
6350:
6343:
6340:
6336:
6331:
6328:
6325:
6289:
6286:
6283:
6278:
6274:
6270:
6265:
6261:
6251:, that is, if
6233:
6230:
6225:
6221:
6170:
6102:
6096:
6091:
6083:
6078:
6074:
6070:
6067:
6062:
6057:
6053:
6047:
6042:
6038:
6030:
6027:
5981:
5977:
5933:
5929:
5902:
5898:
5875:
5871:
5844:
5841:
5836:
5832:
5828:
5825:
5822:
5819:
5816:
5813:
5810:
5805:
5801:
5797:
5794:
5791:
5789:
5785:
5781:
5777:
5776:
5773:
5770:
5767:
5764:
5761:
5756:
5752:
5748:
5745:
5743:
5739:
5735:
5731:
5730:
5713:characteristic
5674:
5671:
5668:
5665:
5662:
5659:
5656:
5651:
5647:
5643:
5640:
5635:
5631:
5627:
5604:
5601:
5597:characteristic
5591:belong to any
5570:
5567:
5562:
5558:
5554:
5551:
5546:
5542:
5538:
5514:
5511:
5508:
5488:
5485:
5482:
5479:
5431:
5426:
5420:
5417:
5412:
5409:
5406:
5378:
5310:
5304:
5295:
5290:
5286:
5280:
5275:
5270:
5266:
5258:
5253:
5250:
5245:
5239:
5236:
5223:
5220:
5216:
5211:
5208:
5165:
5072:
5069:
5066:
5061:
5057:
5053:
5050:
5045:
5041:
5037:
5015:
5009:
5003:
4999:
4991:
4987:
4983:
4977:
4971:
4967:
4959:
4955:
4931:
4925:
4921:
4913:
4909:
4905:
4899:
4893:
4889:
4881:
4877:
4856:
4851:
4845:
4840:
4837:
4834:
4831:
4825:
4820:
4815:
4811:
4807:
4802:
4798:
4777:
4772:
4766:
4761:
4758:
4755:
4752:
4746:
4741:
4737:
4692:
4685:
4680:
4676:
4670:
4665:
4660:
4656:
4648:
4643:
4640:
4635:
4611:
4606:
4602:
4596:
4591:
4586:
4582:
4574:
4569:
4566:
4561:
4539:
4535:
4512:
4508:
4487:
4481:
4475:
4471:
4465:
4459:
4453:
4449:
4420:
4415:
4411:
4405:
4400:
4395:
4391:
4356:
4353:
4350:
4347:
4344:
4341:
4338:
4333:
4329:
4298:
4295:
4275:characteristic
4266:
4263:
4262:
4261:
4250:
4245:
4241:
4235:
4231:
4227:
4224:
4221:
4218:
4213:
4209:
4205:
4202:
4199:
4196:
4193:
4190:
4187:
4184:
4181:
4178:
4173:
4169:
4165:
4162:
4157:
4153:
4149:
4129:
4123:
4120:
4117:
4114:
4111:
4106:
4102:
4098:
4095:
4088:
4084:
4080:
4077:
4072:
4068:
4064:
4061:
4058:
4055:
4052:
4049:
4043:
4038:
4034:
4013:
4007:
4004:
4001:
3998:
3995:
3990:
3986:
3982:
3979:
3974:
3971:
3968:
3965:
3962:
3959:
3953:
3948:
3944:
3940:
3935:
3931:
3910:
3907:
3904:
3901:
3898:
3893:
3889:
3877:
3864:
3859:
3852:
3849:
3845:
3840:
3837:
3833:
3828:
3825:
3822:
3819:
3816:
3813:
3810:
3805:
3801:
3797:
3794:
3789:
3785:
3781:
3761:
3755:
3752:
3748:
3743:
3740:
3735:
3731:
3727:
3722:
3718:
3714:
3709:
3705:
3684:
3681:
3678:
3675:
3672:
3667:
3663:
3649:is zero, then
3638:
3635:
3632:
3629:
3626:
3621:
3617:
3613:
3610:
3605:
3601:
3597:
3560:
3555:
3550:
3543:
3540:
3535:
3532:
3526:
3523:
3519:
3513:
3507:
3503:
3500:
3494:
3491:
3487:
3483:
3480:
3477:
3474:
3471:
3468:
3463:
3459:
3436:
3430:
3427:
3422:
3419:
3413:
3410:
3405:
3401:
3397:
3392:
3388:
3363:
3359:
3356:
3350:
3345:
3341:
3313:
3310:
3307:
3302:
3298:
3294:
3291:
3286:
3282:
3278:
3243:
3240:
3235:
3231:
3227:
3224:
3219:
3215:
3211:
3191:
3188:
3185:
3182:
3179:
3174:
3170:
3150:
3147:
3135:
3132:
3129:
3116:
3109:
3100:
3093:
3083:
3076:
3062:
3055:
3046:
3037:
3028:
3010:
3007:
3004:
2980:
2976:
2970:
2966:
2962:
2957:
2953:
2949:
2944:
2940:
2934:
2930:
2926:
2921:
2917:
2913:
2908:
2904:
2898:
2894:
2890:
2885:
2881:
2877:
2872:
2868:
2864:
2861:
2841:
2838:
2835:
2832:
2829:
2824:
2820:
2816:
2813:
2808:
2804:
2800:
2787:
2780:
2773:
2751:
2750:
2734:
2731:
2728:
2725:
2714:
2699:
2696:
2693:
2690:
2666:
2650:
2647:
2637:
2630:
2623:
2594:
2586:
2582:
2578:
2571:
2567:
2563:
2558:
2554:
2550:
2547:
2544:
2541:
2538:
2535:
2532:
2527:
2523:
2519:
2516:
2513:
2508:
2504:
2500:
2497:
2494:
2491:
2486:
2482:
2478:
2475:
2449:
2445:
2424:
2419:
2415:
2409:
2405:
2400:
2397:
2392:
2388:
2384:
2380:
2377:
2372:
2368:
2362:
2358:
2354:
2351:
2346:
2342:
2337:
2334:
2331:
2328:
2325:
2322:
2318:
2298:
2295:
2292:
2289:
2286:
2281:
2277:
2273:
2270:
2265:
2261:
2257:
2235:
2231:
2225:
2221:
2216:
2213:
2208:
2204:
2199:
2195:
2191:
2171:
2168:
2165:
2162:
2159:
2154:
2150:
2127:
2122:
2118:
2112:
2108:
2104:
2099:
2095:
2091:
2086:
2082:
2076:
2072:
2068:
2063:
2059:
2055:
2050:
2046:
2040:
2036:
2032:
2027:
2023:
2019:
2014:
2010:
1989:
1986:
1983:
1980:
1977:
1974:
1969:
1965:
1961:
1958:
1953:
1949:
1945:
1928:are the three
1924:
1917:
1910:
1883:
1880:
1867:
1864:
1851:
1848:
1845:
1842:
1839:
1836:
1833:
1813:
1807:
1804:
1800:
1795:
1790:
1786:
1782:
1777:
1773:
1750:
1746:
1742:
1737:
1733:
1729:
1724:
1720:
1697:
1693:
1689:
1684:
1680:
1676:
1671:
1667:
1637:
1629:
1625:
1621:
1616:
1611:
1607:
1603:
1600:
1597:
1594:
1591:
1588:
1585:
1580:
1576:
1572:
1566:
1562:
1559:
1556:
1555:
1547:
1543:
1539:
1532:
1528:
1524:
1521:
1518:
1515:
1509:
1505:
1502:
1499:
1498:
1492:
1489:
1485:
1480:
1477:
1474:
1470:
1467:
1464:
1463:
1443:
1440:
1437:
1434:
1431:
1428:
1425:
1422:
1417:
1413:
1369:
1366:
1362:
1357:
1354:
1351:
1348:
1328:
1325:
1322:
1319:
1316:
1313:
1310:
1305:
1301:
1297:
1294:
1289:
1285:
1281:
1254:
1251:
1248:
1245:
1242:
1237:
1233:
1220:
1217:
1179:
1175:
1168:
1164:
1160:
1157:
1154:
1151:
1148:
1145:
1140:
1136:
1132:
1126:
1123:
1116:
1112:
1107:
1104:
1101:
1098:
1095:
1089:
1084:
1080:
1074:
1071:
1044:
1039:
1036:
1031:
1026:
1022:
957:
954:
951:
948:
913:
910:
907:
904:
901:
898:
895:
892:
887:
883:
879:
876:
871:
867:
863:
843:
840:
836:René Descartes
832:François Viète
709:relative error
614:conic sections
487:conic sections
430:
427:
418:characteristic
410:
409:
395:
390:
325:cubic function
301:
298:
295:
292:
289:
286:
283:
278:
274:
270:
267:
262:
258:
254:
239:cubic equation
214:
211:
208:
205:
202:
199:
196:
193:
190:
187:
184:
181:
178:
175:
172:
167:
164:
159:
156:
154:
152:
148:
144:
141:
138:
135:
132:
127:
123:
119:
116:
111:
107:
102:
96:
93:
88:
85:
83:
81:
78:
75:
72:
69:
68:
32:Cubic function
15:
9:
6:
4:
3:
2:
16176:
16165:
16162:
16160:
16157:
16155:
16152:
16151:
16149:
16134:
16133:Gröbner basis
16131:
16129:
16126:
16124:
16121:
16119:
16116:
16114:
16111:
16109:
16106:
16104:
16101:
16099:
16098:Factorization
16096:
16095:
16093:
16089:
16083:
16080:
16078:
16075:
16073:
16070:
16068:
16065:
16063:
16060:
16058:
16055:
16053:
16050:
16048:
16045:
16043:
16040:
16038:
16035:
16034:
16032:
16030:By properties
16028:
16022:
16019:
16017:
16014:
16012:
16009:
16005:
16002:
16001:
16000:
15997:
15993:
15990:
15989:
15988:
15985:
15981:
15978:
15977:
15976:
15973:
15969:
15966:
15965:
15964:
15961:
15959:
15956:
15954:
15951:
15950:
15948:
15946:
15941:
15937:
15933:
15926:
15921:
15919:
15914:
15912:
15907:
15906:
15903:
15897:
15893:
15889:
15885:
15882:
15879:
15875:
15872:
15868:
15864:
15863:
15858:
15854:
15853:
15844:
15840:
15836:
15832:
15828:
15824:
15820:
15816:
15812:
15807:
15804:
15800:
15796:
15792:
15788:
15784:
15780:
15776:
15772:
15767:
15764:
15758:
15754:
15750:
15745:
15742:
15738:
15734:
15730:
15726:
15722:
15718:
15714:
15710:
15705:
15702:
15698:
15694:
15690:
15686:
15682:
15678:
15674:
15670:
15665:
15662:
15656:
15652:
15651:Basic algebra
15648:
15644:
15641:
15637:
15633:
15629:
15625:
15621:
15617:
15613:
15609:
15605:
15601:
15596:
15593:
15589:
15585:
15581:
15577:
15573:
15569:
15565:
15561:
15557:
15553:
15548:
15544:
15538:
15534:
15530:
15525:
15524:
15515:
15511:
15507:
15503:
15499:
15495:
15490:
15489:
15479:
15478:3-540-43826-2
15475:
15471:
15467:
15463:
15457:
15455:
15448:
15443:
15437:
15433:
15432:
15424:
15418:
15413:
15407:
15403:
15402:
15394:
15386:
15382:
15378:
15372:
15365:
15363:3-540-13610-X
15359:
15355:
15351:
15347:
15341:
15334:
15330:
15326:
15322:
15318:
15314:
15307:
15300:
15296:
15292:
15288:
15284:
15280:
15279:
15271:
15264:
15260:
15256:
15252:
15248:
15244:
15243:
15235:
15227:
15223:
15219:
15215:
15211:
15207:
15203:
15199:
15195:
15191:
15184:
15177:
15166:
15162:
15158:
15154:
15150:
15146:
15142:
15135:
15128:
15121:
15115:
15108:
15107:
15100:
15093:
15087:
15079:
15077:0-87819-622-6
15073:
15070:. CRC Press.
15069:
15068:
15060:
15052:
15048:
15044:
15040:
15036:
15032:
15031:
15023:
15009:
15008:Math solution
15005:
14999:
14992:
14988:
14986:9781974130924
14982:
14978:
14977:
14969:
14961:
14957:
14953:
14949:
14945:
14941:
14940:
14932:
14925:
14923:
14921:
14913:
14909:
14905:
14901:
14897:
14893:
14892:
14887:
14880:
14872:
14870:9780321016188
14866:
14862:
14858:
14857:
14849:
14842:
14838:
14837:
14832:
14828:
14821:
14814:
14810:
14806:
14802:
14798:
14794:
14787:
14780:
14776:
14775:
14770:
14766:
14759:
14752:
14750:81-86050-86-8
14746:
14742:
14741:
14736:
14730:
14722:
14716:
14712:
14711:
14703:
14696:
14691:
14684:
14680:
14674:
14665:
14658:
14652:
14648:
14644:
14638:
14630:
14624:
14620:
14616:
14610:
14602:
14596:
14592:
14588:
14587:
14582:
14576:
14569:
14564:
14562:
14554:
14549:
14541:
14535:
14531:
14530:
14522:
14514:
14508:
14504:
14499:
14498:
14489:
14481:
14475:
14471:
14470:
14462:
14456:
14455:0-387-12159-5
14452:
14446:
14444:
14435:
14429:
14425:
14424:
14416:
14414:
14406:
14400:
14396:
14392:
14388:
14384:
14377:
14373:
14360:
14351:
14343:
14334:
14326:
14310:
14302:
14298:
14294:
14283:
14276:
14273:. Textually:
14272:
14268:
14263:
14259:
14255:
14251:
14248:one may read
14246:
14242:
14241:
14236:
14232:
14224:
14220:
14209:
14206:
14203:
14200:
14199:Rayleigh wave
14196:
14193:
14189:
14186:
14183:
14179:
14175:
14171:
14168:
14164:
14160:
14156:
14155:
14146:
14142:
14139:
14136:
14132:
14129:
14126:
14122:
14119:
14115:
14111:
14107:
14103:
14100:
14096:
14092:
14088:
14085:
14081:
14077:
14074:
14070:
14055:
14051:
14047:
14044:
14036:
14032:
14028:
14023:
14020:of a regular
14019:
14015:
14012:
14008:
14007:complex plane
14004:
14000:
13996:
13993:
13990:
13986:
13982:
13978:
13975:
13974:
13968:
13960:
13953:
13945:
13934:
13924:
13916:
13905:
13895:
13887:
13876:
13870:
13863:
13857:
13850:
13840:
13819:
13814:
13810:
13806:
13803:
13798:
13794:
13788:
13784:
13780:
13777:
13772:
13767:
13763:
13759:
13756:
13754:
13749:
13742:
13737:
13733:
13729:
13726:
13721:
13716:
13712:
13708:
13706:
13701:
13689:
13685:
13657:
13652:
13648:
13642:
13638:
13632:
13628:
13624:
13621:
13613:
13608:
13604:
13598:
13594:
13590:
13585:
13580:
13576:
13570:
13566:
13562:
13557:
13552:
13548:
13542:
13538:
13534:
13529:
13525:
13519:
13514:
13510:
13506:
13501:
13497:
13491:
13486:
13482:
13478:
13473:
13469:
13463:
13458:
13454:
13447:
13444:
13436:
13431:
13427:
13423:
13418:
13413:
13409:
13405:
13400:
13395:
13391:
13384:
13381:
13376:
13371:
13367:
13363:
13358:
13353:
13349:
13345:
13343:
13338:
13331:
13323:
13319:
13313:
13309:
13305:
13300:
13296:
13290:
13286:
13282:
13277:
13273:
13267:
13263:
13256:
13251:
13246:
13242:
13238:
13233:
13228:
13224:
13220:
13215:
13210:
13206:
13202:
13197:
13193:
13187:
13183:
13179:
13177:
13172:
13158:
13154:
13147:
13129:
13114:
13111:
13108:
13105:
13100:
13096:
13075:
13072:
13069:
13064:
13060:
13039:
13036:
13012:
13009:
13003:
13000:
12994:
12991:
12988:
12985:
12965:
12962:
12959:
12956:
12951:
12947:
12926:
12923:
12920:
12917:
12914:
12909:
12905:
12899:
12895:
12869:
12865:
12861:
12856:
12852:
12842:
12839:
12833:
12828:
12824:
12814:
12801:
12798:
12772:
12768:
12745:
12741:
12720:
12717:
12712:
12708:
12704:
12701:
12698:
12695:
12690:
12686:
12663:
12658:
12654:
12631:
12626:
12622:
12599:
12594:
12590:
12586:
12581:
12576:
12572:
12568:
12565:
12545:
12540:
12536:
12530:
12526:
12522:
12519:
12510:
12489:
12480:
12471:
12462:
12453:
12444:
12439:
12431:
12422:
12412:
12404:
12390:
12385:
12380:
12363:
12355:
12351:
12345:
12341:
12337:
12332:
12328:
12324:
12321:
12316:
12312:
12302:
12299:
12293:
12291:
12284:
12280:
12272:
12264:
12260:
12256:
12253:
12248:
12244:
12238:
12234:
12230:
12225:
12221:
12211:
12208:
12202:
12200:
12193:
12189:
12181:
12173:
12169:
12165:
12160:
12156:
12152:
12147:
12143:
12133:
12130:
12124:
12122:
12115:
12111:
12094:
12085:
12076:
12055:
12050:
12046:
12042:
12039:
12034:
12030:
12024:
12020:
12016:
12011:
12007:
12003:
12001:
11994:
11990:
11982:
11977:
11973:
11967:
11963:
11959:
11954:
11950:
11946:
11943:
11938:
11934:
11930:
11928:
11921:
11917:
11909:
11904:
11900:
11896:
11891:
11887:
11883:
11878:
11874:
11870:
11868:
11861:
11857:
11840:
11831:
11822:
11804:
11799:
11795:
11791:
11788:
11785:
11781:
11777:
11772:
11766:
11761:
11758:
11755:
11752:
11746:
11743:
11734:
11728:
11724:
11717:
11712:
11704:
11699:
11695:
11691:
11687:
11680:
11676:
11672:
11668:
11662:
11660:
11656:
11652:
11648:
11644:
11641:In his paper
11634:
11626:
11622:
11615:
11604:
11600:
11591:
11579:
11575:
11557:
11551:
11548:
11542:
11538:
11532:
11515:
11506:
11495:
11482:
11473:
11462:
11449:
11440:
11429:
11420:
11412:
11403:
11394:
11373:
11368:
11364:
11358:
11353:
11348:
11344:
11336:
11331:
11328:
11323:
11320:
11317:
11308:
11295:
11292:
11287:
11282:
11278:
11272:
11264:
11260:
11253:
11250:
11245:
11235:
11231:
11214:
11208:
11195:
11192:
11184:
11180:
11176:
11170:
11166:
11160:
11157:
11154:
11149:
11145:
11131:
11122:
11114:
11110:
11104:
11097:
11088:
11080:
11076:
11069:
11065:
11061:
11055:
11053:
11049:
11039:
11026:
11021:
11015:
11012:
11007:
11004:
11001:
10990:
10985:
10983:
10967:
10961:
10958:
10953:
10950:
10944:
10941:
10921:
10916:
10913:
10908:
10905:
10902:
10899:
10891:
10887:
10883:
10867:
10864:
10861:
10856:
10852:
10848:
10845:
10840:
10836:
10832:
10812:
10809:
10806:
10801:
10797:
10793:
10790:
10785:
10781:
10777:
10768:
10751:
10742:
10737:
10733:
10727:
10722:
10717:
10713:
10705:
10700:
10697:
10692:
10686:
10680:
10671:
10666:
10662:
10656:
10651:
10646:
10642:
10634:
10629:
10626:
10621:
10615:
10612:
10592:
10589:
10586:
10583:
10580:
10577:
10557:
10551:
10542:
10537:
10533:
10527:
10522:
10517:
10513:
10505:
10500:
10497:
10492:
10486:
10483:
10464:
10458:
10449:
10444:
10440:
10434:
10429:
10424:
10420:
10412:
10407:
10404:
10399:
10393:
10390:
10370:
10346:
10339:
10334:
10330:
10324:
10319:
10314:
10310:
10302:
10297:
10294:
10289:
10267:
10261:
10257:
10253:
10247:
10242:
10238:
10234:
10211:
10208:
10203:
10198:
10194:
10188:
10185:
10182:
10179:
10174:
10170:
10143:
10135:
10132:
10126:
10123:
10115:
10111:
10107:
10102:
10098:
10091:
10086:
10082:
10078:
10076:
10066:
10062:
10056:
10052:
10048:
10045:
10037:
10033:
10029:
10024:
10020:
10013:
10008:
10004:
10000:
9998:
9985:
9981:
9977:
9974:
9963:
9959:
9955:
9952:
9946:
9944:
9939:
9928:
9912:
9907:
9903:
9893:
9887:
9870:
9865:
9862:
9857:
9854:
9852:
9847:
9844:
9837:
9834:
9831:
9829:
9822:
9818:
9814:
9809:
9805:
9780:
9777:
9774:
9771:
9768:
9765:
9762:
9753:
9740:
9737:
9734:
9731:
9725:
9722:
9719:
9710:
9707:
9704:
9701:
9698:
9692:
9687:
9683:
9679:
9674:
9670:
9649:
9646:
9643:
9640:
9637:
9617:
9603:
9599:
9595:
9589:
9587:
9586:
9581:
9577:
9573:
9563:
9561:
9551:
9543:
9539:
9535:
9531:
9522:
9513:
9508:
9503:
9497:
9487:
9478:
9421:
9412:
9403:
9382:
9374:
9370:
9366:
9361:
9357:
9345:
9341:
9337:
9332:
9328:
9316:
9312:
9308:
9303:
9299:
9290:
9286:
9282:
9262:
9256:
9247:
9242:
9230:
9222:
9218:
9211:over a field
9210:
9200:
9138:
9134:
9130:
9124:
9115:
9110:
9102:
9095:
9090:
9086:
9077:
9073:
9069:
9059:
9051:
9042:
9030:
9015:
9010:
9004:
9000:
8994:
8983:One real root
8980:
8979:sum to zero.
8957:
8948:
8940:
8936:
8926:
8917:
8909:
8905:
8896:
8893:
8890:
8885:
8867:
8864:
8859:
8854:
8844:
8820:
8783:
8776:
8772:
8769:
8759:
8756:
8751:
8748:
8741:
8737:
8734:
8725:
8712:
8708:
8705:
8702:
8699:
8696:
8693:
8690:
8679:
8673:
8669:
8666:
8660:
8657:
8653:
8646:
8642:
8639:
8629:
8626:
8621:
8618:
8611:
8607:
8604:
8599:
8596:
8590:
8586:
8583:
8576:
8573:
8568:
8563:
8560:
8555:
8551:
8542:
8538:
8534:
8530:
8525:
8522:
8519:
8511:
8503:
8483:
8480:
8477:
8472:
8458:
8456:
8452:
8448:
8444:
8439:
8437:
8433:
8423:
8420:
8409:
8401:
8393:
8389:
8385:
8379:
8362:
8358:
8354:
8351:
8344:
8340:
8336:
8330:
8326:
8323:
8316:
8312:
8306:
8302:
8290:
8286:
8280:
8266:
8261:
8257:
8253:
8248:
8242:
8236:
8232:
8229:
8225:
8206:
8199:
8193:
8179:
8175:
8168:
8161:
8154:
8147:
8142:
8136:
8131:
8123:
8116:
8110:
8106:
8085:
8082:
8079:
8066:
8061:
8054:
8051:
8042:
8039:
8034:
8031:
8024:
8020:
8017:
8012:
8009:
8003:
7999:
7996:
7990:
7987:
7981:
7978:
7975:
7973:
7966:
7962:
7954:
7951:
7948:
7945:
7931:
7928:
7923:
7919:
7915:
7912:
7907:
7903:
7899:
7886:
7881:
7874:
7870:
7867:
7857:
7854:
7844:
7836:
7833:
7826:
7822:
7819:
7814:
7811:
7805:
7801:
7798:
7791:
7788:
7783:
7776:
7767:
7756:
7753:
7750:
7748:
7741:
7737:
7724:
7718:
7707:
7694:
7691:
7688:
7685:
7682:
7679:
7676:
7666:
7660:
7656:
7653:
7650:
7644:
7640:
7633:
7629:
7626:
7616:
7613:
7608:
7605:
7598:
7594:
7591:
7586:
7583:
7577:
7573:
7570:
7562:
7559:
7554:
7548:
7545:
7540:
7536:
7515:
7508:
7504:
7501:
7491:
7488:
7483:
7480:
7474:
7468:
7465:
7459:
7456:
7436:
7433:
7427:
7423:
7420:
7409:
7406:
7401:
7398:
7392:
7389:
7386:
7383:
7380:
7377:
7374:
7371:
7366:
7362:
7358:
7338:
7333:
7328:
7324:
7301:
7293:
7290:
7285:
7279:
7276:
7273:
7253:
7250:
7244:
7241:
7235:
7232:
7229:
7226:
7223:
7220:
7217:
7214:
7211:
7208:
7203:
7199:
7195:
7180:
7176:
7172:
7166:, let us set
7163:
7159:
7155:
7149:
7147:
7142:
7138:
7121:
7118:
7113:
7109:
7105:
7102:
7097:
7093:
7089:
7081:
7076:
7063:
7060:
7057:
7054:
7051:
7048:
7045:
7035:
7028:
7024:
7021:
7018:
7012:
7008:
7000:
6996:
6993:
6983:
6980:
6975:
6972:
6965:
6961:
6958:
6953:
6950:
6943:
6939:
6936:
6928:
6925:
6920:
6914:
6911:
6906:
6902:
6881:
6878:
6875:
6872:
6869:
6866:
6863:
6858:
6854:
6846:
6842:
6838:
6834:
6830:
6829:
6824:
6823:Galois theory
6809:
6796:
6793:
6788:
6780:
6775:
6749:
6744:
6733:
6730:
6726:
6721:
6718:
6712:
6691:
6685:
6682:
6677:
6674:
6651:
6648:
6645:
6640:
6632:
6627:
6613:
6588:
6563:
6560:
6557:
6554:
6551:
6545:
6542:
6538:
6534:
6527:
6522:
6518:
6511:
6501:
6498:
6493:
6489:
6485:
6482:
6478:
6471:
6468:
6464:
6459:
6456:
6451:
6447:
6414:
6401:
6382:
6376:
6371:
6361:
6358:
6355:
6352:
6348:
6341:
6338:
6334:
6329:
6326:
6323:
6287:
6284:
6281:
6276:
6268:
6263:
6248:
6231:
6228:
6223:
6208:
6191:
6187:
6168:
6100:
6094:
6089:
6081:
6076:
6068:
6065:
6060:
6055:
6045:
6040:
6028:
6025:
6016:
6009:
5979:
5961:
5931:
5918:
5900:
5873:
5859:
5842:
5839:
5834:
5830:
5826:
5823:
5820:
5817:
5814:
5811:
5808:
5803:
5799:
5795:
5792:
5790:
5783:
5771:
5768:
5765:
5762:
5759:
5754:
5750:
5746:
5744:
5737:
5719:
5716:
5714:
5710:
5705:
5701:
5697:
5693:
5688:
5672:
5669:
5666:
5663:
5660:
5657:
5654:
5649:
5645:
5641:
5638:
5633:
5629:
5625:
5615:
5610:
5609:cubic formula
5600:
5598:
5594:
5581:
5568:
5565:
5560:
5556:
5552:
5549:
5544:
5540:
5536:
5528:
5512:
5509:
5506:
5486:
5483:
5480:
5477:
5467:
5460:
5453:
5449:
5443:
5429:
5424:
5418:
5415:
5410:
5407:
5404:
5376:
5308:
5302:
5293:
5288:
5284:
5278:
5273:
5268:
5264:
5256:
5251:
5248:
5243:
5237:
5234:
5221:
5218:
5214:
5209:
5206:
5196:
5189:
5187:
5182:
5163:
5107:
5102:
5100:
5096:
5095:
5090:
5086:
5085:Galois theory
5070:
5067:
5064:
5059:
5055:
5051:
5048:
5043:
5039:
5035:
5026:
5013:
5007:
5001:
4997:
4989:
4985:
4981:
4975:
4969:
4965:
4957:
4953:
4929:
4923:
4919:
4911:
4907:
4903:
4897:
4891:
4887:
4879:
4875:
4854:
4849:
4843:
4838:
4835:
4832:
4829:
4823:
4818:
4813:
4809:
4805:
4800:
4796:
4775:
4770:
4764:
4759:
4756:
4753:
4750:
4744:
4739:
4735:
4727:
4723:
4719:
4715:
4710:
4708:
4703:
4690:
4683:
4678:
4674:
4668:
4663:
4658:
4654:
4646:
4641:
4638:
4633:
4609:
4604:
4600:
4594:
4589:
4584:
4580:
4572:
4567:
4564:
4559:
4537:
4533:
4510:
4506:
4485:
4479:
4473:
4469:
4463:
4457:
4451:
4447:
4436:
4418:
4413:
4409:
4403:
4398:
4393:
4389:
4378:
4354:
4351:
4348:
4345:
4342:
4339:
4336:
4331:
4327:
4317:
4315:
4311:
4307:
4303:
4294:
4290:
4288:
4284:
4278:
4276:
4272:
4248:
4243:
4233:
4229:
4225:
4222:
4211:
4207:
4203:
4200:
4194:
4191:
4188:
4185:
4182:
4179:
4176:
4171:
4167:
4163:
4160:
4155:
4151:
4147:
4127:
4118:
4115:
4112:
4109:
4104:
4100:
4093:
4086:
4082:
4078:
4075:
4070:
4066:
4062:
4059:
4056:
4053:
4050:
4047:
4041:
4036:
4032:
4011:
4002:
3999:
3996:
3993:
3988:
3984:
3977:
3972:
3969:
3966:
3963:
3960:
3957:
3951:
3946:
3942:
3938:
3933:
3929:
3908:
3905:
3902:
3899:
3896:
3891:
3887:
3878:
3862:
3857:
3850:
3847:
3843:
3838:
3835:
3831:
3826:
3823:
3820:
3817:
3814:
3811:
3808:
3803:
3799:
3795:
3792:
3787:
3783:
3779:
3759:
3753:
3750:
3746:
3741:
3738:
3733:
3729:
3725:
3720:
3716:
3712:
3707:
3703:
3682:
3679:
3676:
3673:
3670:
3665:
3661:
3652:
3651:
3650:
3636:
3633:
3630:
3627:
3624:
3619:
3615:
3611:
3608:
3603:
3599:
3595:
3587:
3584:By using the
3582:
3580:
3576:
3571:
3558:
3553:
3548:
3541:
3538:
3533:
3530:
3524:
3521:
3517:
3511:
3505:
3501:
3498:
3492:
3489:
3485:
3481:
3478:
3475:
3472:
3469:
3466:
3461:
3457:
3447:
3434:
3428:
3425:
3420:
3417:
3411:
3408:
3403:
3399:
3395:
3390:
3386:
3376:
3361:
3357:
3354:
3348:
3343:
3339:
3328:
3311:
3308:
3305:
3300:
3296:
3292:
3289:
3284:
3280:
3276:
3266:
3262:
3241:
3238:
3233:
3229:
3225:
3222:
3217:
3213:
3209:
3189:
3186:
3183:
3180:
3177:
3172:
3168:
3158:
3156:
3155:multiple root
3149:Multiple root
3146:
3133:
3130:
3115:
3108:
3099:
3092:
3082:
3075:
3069:
3061:
3054:
3045:
3036:
3027:
3021:
3008:
3005:
2993:
2978:
2968:
2964:
2960:
2955:
2951:
2942:
2932:
2928:
2924:
2919:
2915:
2906:
2896:
2892:
2888:
2883:
2879:
2870:
2866:
2862:
2839:
2836:
2833:
2830:
2827:
2822:
2818:
2814:
2811:
2806:
2802:
2798:
2786:
2779:
2772:
2766:
2764:
2760:
2748:
2732:
2729:
2726:
2715:
2713:
2697:
2694:
2691:
2680:
2679:
2678:
2656:
2646:
2636:
2629:
2622:
2617:
2612:
2610:
2605:
2592:
2584:
2580:
2576:
2569:
2561:
2556:
2552:
2548:
2545:
2542:
2539:
2536:
2533:
2530:
2525:
2521:
2517:
2511:
2506:
2498:
2495:
2492:
2489:
2484:
2480:
2473:
2447:
2443:
2422:
2417:
2413:
2407:
2403:
2398:
2395:
2390:
2386:
2382:
2378:
2375:
2370:
2366:
2360:
2356:
2352:
2349:
2344:
2340:
2335:
2332:
2329:
2326:
2323:
2320:
2316:
2296:
2293:
2290:
2287:
2284:
2279:
2275:
2271:
2268:
2263:
2259:
2255:
2246:
2233:
2229:
2223:
2219:
2214:
2211:
2206:
2202:
2197:
2193:
2189:
2169:
2166:
2163:
2160:
2157:
2152:
2148:
2138:
2125:
2120:
2110:
2106:
2102:
2097:
2093:
2084:
2074:
2070:
2066:
2061:
2057:
2048:
2038:
2034:
2030:
2025:
2021:
2012:
2008:
1987:
1984:
1981:
1978:
1975:
1972:
1967:
1963:
1959:
1956:
1951:
1947:
1943:
1935:
1931:
1923:
1916:
1909:
1903:
1901:
1897:
1896:multiple root
1893:
1889:
1879:
1877:
1873:
1863:
1849:
1846:
1843:
1840:
1837:
1834:
1831:
1811:
1805:
1802:
1798:
1793:
1788:
1784:
1780:
1775:
1771:
1748:
1744:
1740:
1735:
1731:
1727:
1722:
1718:
1695:
1691:
1687:
1682:
1678:
1674:
1669:
1665:
1657:
1652:
1635:
1627:
1623:
1619:
1614:
1609:
1605:
1601:
1598:
1595:
1592:
1589:
1586:
1583:
1578:
1574:
1570:
1560:
1557:
1545:
1541:
1537:
1530:
1526:
1522:
1519:
1516:
1513:
1503:
1500:
1490:
1487:
1483:
1478:
1475:
1468:
1465:
1441:
1438:
1435:
1432:
1429:
1426:
1423:
1420:
1415:
1411:
1403:
1400:one gets the
1394:
1391:
1367:
1364:
1360:
1355:
1352:
1349:
1346:
1326:
1323:
1320:
1317:
1314:
1311:
1308:
1303:
1299:
1295:
1292:
1287:
1283:
1279:
1270:
1268:
1252:
1249:
1246:
1243:
1240:
1235:
1231:
1216:
1214:
1210:
1205:
1202:
1198:
1177:
1173:
1166:
1162:
1158:
1155:
1152:
1149:
1146:
1143:
1138:
1134:
1130:
1124:
1121:
1114:
1110:
1105:
1102:
1099:
1096:
1093:
1087:
1082:
1078:
1072:
1069:
1059:
1042:
1037:
1034:
1029:
1024:
1020:
1009:
983:
979:
955:
952:
949:
946:
937:
935:
931:
930:Gauss's lemma
927:
911:
908:
905:
902:
899:
896:
893:
890:
885:
881:
877:
874:
869:
865:
861:
853:
849:
842:Factorization
839:
837:
833:
829:
827:
823:
819:
814:
812:
806:
804:
803:
798:
793:
790:
786:
782:
776:
772:
768:
762:
761:Zuanne da Coi
758:
749:
745:
743:
730:
726:
722:
717:
712:
711:of about 10.
710:
706:
703:. Writing in
700:
696:
692:
687:
683:
678:
676:
672:
668:
664:
660:
659:
654:
650:
646:
642:
636:
632:
628:
621:
617:
615:
611:
610:
605:
600:
596:
589:
585:
579:
575:
571:
567:
562:
558:
557:Jigu Suanjing
554:
553:Wang Xiaotong
550:
542:
538:
534:
530:
526:
522:
518:
514:
510:
506:
502:
500:
496:
492:
488:
484:
480:
476:
472:
469:
464:
462:
458:
454:
453:
448:
444:
440:
436:
426:
424:
419:
415:
407:
403:
399:
396:
394:
391:
388:
384:
380:
376:
372:
368:
364:
363:cubic formula
360:
359:algebraically
357:
356:
355:
353:
349:
330:
326:
322:
317:
316:is not zero.
299:
296:
293:
290:
287:
284:
281:
276:
272:
268:
265:
260:
256:
252:
244:
240:
236:
209:
206:
203:
194:
191:
188:
179:
176:
173:
165:
162:
157:
155:
146:
142:
139:
136:
133:
130:
125:
121:
117:
114:
109:
105:
100:
94:
91:
86:
84:
76:
70:
58:
52:
47:
44:
39:
33:
26:
22:
16128:Discriminant
16047:Multivariate
15991:
15860:
15814:
15810:
15774:
15770:
15752:
15712:
15708:
15672:
15668:
15650:
15603:
15599:
15555:
15551:
15532:
15497:
15493:
15469:
15430:
15423:
15417:§6.2, p. 134
15400:
15393:
15384:
15371:
15349:
15340:
15316:
15312:
15306:
15282:
15276:
15270:
15246:
15240:
15234:
15193:
15189:
15176:
15165:the original
15144:
15140:
15127:
15119:
15114:
15104:
15099:
15086:
15066:
15059:
15034:
15028:
15022:
15011:. Retrieved
15007:
14998:
14990:
14975:
14968:
14943:
14937:
14898:(1): 12–21,
14895:
14889:
14886:Mazur, Barry
14879:
14855:
14848:
14834:
14820:
14796:
14792:
14786:
14772:
14758:
14739:
14729:
14713:. Springer.
14709:
14702:
14690:
14679:Omar Khayyam
14673:
14664:
14646:
14637:
14618:
14609:
14585:
14575:
14548:
14528:
14521:
14496:
14488:
14468:
14461:
14422:
14382:
14376:
14349:
14341:
14332:
14324:
14308:
14300:
14296:
14292:
14282:
14274:
14266:
14261:
14257:
14253:
14249:
14238:
14223:
14125:Bézier curve
14110:Cauchy–Euler
14031:circumradius
13966:
13963:Applications
13951:
13943:
13932:
13922:
13914:
13903:
13893:
13885:
13874:
13868:
13861:
13855:
13848:
13838:
13156:
13152:
13145:
13142:
12815:
12787:in place of
12511:
12487:
12478:
12469:
12460:
12451:
12442:
12429:
12420:
12410:
12402:
12388:
12381:
12092:
12083:
12074:
11838:
11829:
11820:
11726:
11722:
11715:
11700:
11693:
11689:
11685:
11678:
11674:
11670:
11666:
11663:
11642:
11640:
11624:
11620:
11613:
11602:
11598:
11589:
11577:
11573:
11513:
11504:
11493:
11480:
11471:
11460:
11447:
11438:
11427:
11410:
11401:
11392:
11309:
11212:
11209:
11129:
11120:
11112:
11108:
11105:
11095:
11086:
11078:
11074:
11067:
11063:
11059:
11056:
11045:
10991:, which are
10986:
10889:
10888:in his book
10769:
9891:
9888:
9754:
9601:
9597:
9593:
9590:
9584:
9569:
9557:
9541:
9537:
9533:
9529:
9520:
9511:
9501:
9498:
9485:
9476:
9419:
9410:
9401:
9263:
9254:
9245:
9221:Galois group
9206:
9203:Galois group
9125:
9121:
9101:tangent line
9088:
9084:
9065:
9057:
9049:
9040:
9028:
9013:
9008:
9002:
8998:
8955:
8946:
8938:
8934:
8924:
8915:
8907:
8903:
8894:
8888:
8886:
8842:
8818:
8726:
8543:
8536:
8532:
8528:
8523:
8517:
8515:
8481:
8475:
8440:
8429:
8418:
8407:
8399:
8391:
8387:
8383:
8377:
8288:
8284:
8281:
8265:line segment
8259:
8255:
8251:
8247:Omar Khayyám
8240:
8234:
8230:
8227:
8223:
8220:
8204:
8197:
8173:
8166:
8159:
8152:
8145:
8134:
8129:
8121:
8114:
8111:
8104:
7716:
7713:
7351:This gives
7178:
7174:
7170:
7161:
7157:
7153:
7150:
7143:
7136:
7077:
6826:
6820:
6614:
6586:
6402:
6246:
6206:
6189:
6185:
6017:
6007:
5959:
5860:
5720:
5717:
5703:
5699:
5695:
5691:
5613:
5608:
5606:
5582:
5465:
5458:
5451:
5447:
5444:
5194:
5190:
5105:
5103:
5098:
5092:
5027:
4712:As shown in
4711:
4704:
4435:discriminant
4377:real numbers
4318:
4300:
4291:
4279:
4268:
3583:
3572:
3448:
3377:
3326:
3264:
3260:
3159:
3152:
3113:
3106:
3097:
3090:
3080:
3073:
3059:
3052:
3043:
3034:
3025:
3022:
2994:
2784:
2777:
2770:
2767:
2752:
2655:real numbers
2652:
2634:
2627:
2620:
2613:
2606:
2247:
2139:
1921:
1914:
1907:
1904:
1888:discriminant
1885:
1882:Discriminant
1876:discriminant
1869:
1653:
1401:
1395:
1389:
1271:
1222:
1206:
1204:is a root.)
1200:
1196:
1010:
938:
845:
830:
821:
815:
807:
800:
794:
788:
784:
780:
774:
770:
766:
754:
728:
724:
720:
713:
698:
694:
690:
681:
680:In his book
679:
675:discriminant
656:
652:
649:Al-Muʿādalāt
648:
634:
630:
626:
623:
619:
607:
604:Omar Khayyam
601:
594:
587:
583:
577:
573:
569:
565:
549:Tang dynasty
546:
540:
536:
532:
528:
524:
520:
516:
512:
465:
450:
432:
411:
371:square roots
362:
348:real numbers
329:coefficients
318:
245:of the form
238:
232:
50:
20:
16164:Polynomials
16077:Homogeneous
16072:Square-free
16067:Irreducible
15932:Polynomials
15821:: 264–268,
15781:: 268–276,
15679:: 514–516,
15610:: 347–348,
15562:: 403–408,
15500:(4): 8–12,
15466:Ragni Piene
15037:: 264–268.
14831:"Fibonacci"
14091:eigenvalues
9092:, then the
8139:, the same
6300:a fraction
5527:irreducible
3653:either, if
3202:is zero if
1900:square-free
561:numerically
491:T. L. Heath
443:Hippocrates
16148:Categories
16037:Univariate
15892:Mathologer
15486:References
15013:2022-11-23
14615:Archimedes
14387:Birkhäuser
14369:References
14194:are cubic.
11735:, one has
11419:cube roots
9630:such that
8901:such that
7177:cos
6841:arccosines
6415:, that is
5917:resultants
5101:in Latin.
5097:, meaning
4379:such that
1892:polynomial
483:Archimedes
479:Menaechmus
471:Diophantus
435:Babylonian
375:cube roots
16159:Equations
16123:Resultant
16062:Trinomial
16042:Bivariate
15890:video by
15867:EMS Press
15843:125986006
15835:0025-5572
15803:125870578
15795:0025-5572
15741:126286653
15733:0025-5572
15701:124710259
15693:0025-5572
15640:125643035
15624:0025-5572
15592:125196796
15576:0025-5572
15226:172730765
15210:0025-5572
15051:125986006
14960:124980170
14912:189888034
14093:of a 3×3
14048:π
13778:−
13727:−
13445:−
13257:−
12995:−
12918:−
12696:−
12342:ξ
12325:ξ
12257:ξ
12235:ξ
12043:ξ
12021:ξ
11964:ξ
11947:ξ
11792:π
11759:±
11753:−
11744:ξ
11616:= 1, 2, 3
11533:−
11337:±
11324:−
11273:−
11161:−
11013:−
11008:±
11002:−
10951:−
10934:that is,
10909:−
10890:L'Algebra
10706:−
10693:−
10622:−
10506:−
10493:−
10400:−
10303:±
10290:−
10232:Δ
10189:−
10092:−
10014:−
9978:−
9956:−
9858:−
9835:−
9585:Ars Magna
9576:Tartaglia
9460:Δ
9436:Δ
9367:−
9338:−
9309:−
9278:Δ
9231:that fix
9094:real part
8860:−
8770:−
8738:
8670:π
8658:−
8640:−
8608:
8587:
8569:−
8453:, set by
8434:, and an
8352:−
8021:
8000:
7979:−
7868:−
7834:−
7823:
7802:
7784:−
7754:−
7673:for
7654:π
7645:−
7627:−
7595:
7574:
7555:−
7502:−
7469:θ
7460:
7421:−
7393:−
7390:θ
7387:
7378:−
7375:θ
7372:
7286:−
7245:θ
7236:
7230:−
7227:θ
7224:
7215:−
7212:θ
7209:
7042:for
7022:π
7013:−
6994:−
6962:
6940:
6921:−
6785:Δ
6772:Δ
6675:−
6637:Δ
6624:Δ
6546:∈
6519:ξ
6508:Δ
6490:ξ
6460:−
6368:Δ
6330:−
6273:Δ
6260:Δ
6220:Δ
6073:Δ
6066:−
6052:Δ
6046:±
6037:Δ
6018:Then let
5976:Δ
5928:Δ
5897:Δ
5870:Δ
5809:−
5780:Δ
5760:−
5734:Δ
5507:−
5416:−
5411:±
5405:−
5244:−
5210:−
4986:ε
4954:ε
4908:ε
4876:ε
4836:−
4830:−
4810:ε
4797:ε
4751:−
4736:ε
4647:−
4634:−
4560:−
4226:−
4204:−
4140:and thus
4110:−
4079:−
4060:−
3994:−
3967:−
3897:≠
3742:−
3493:−
3412:−
3128:Δ
3003:Δ
2961:−
2925:−
2889:−
2860:Δ
2724:Δ
2689:Δ
2665:Δ
2531:−
2512:−
2490:−
2396:−
2376:−
2333:−
2190:−
2103:−
2067:−
2031:−
1794:−
1584:−
1523:−
1356:−
953:−
926:reducible
822:Ars Magna
802:Ars Magna
755:In 1535,
686:Fibonacci
379:quadratic
312:in which
177:−
140:−
131:−
16108:Division
16057:Binomial
16052:Monomial
15649:(2009),
14027:inradius
14022:heptagon
13985:geometry
13052:we have
10825:but, if
9588:(1545).
9076:abscissa
8443:rational
8143:denoted
8073:if
7893:if
809:student
404:such as
243:equation
15888:YouTube
15869:, 2001
15632:3620218
15606:(483),
15584:3619617
15558:(492),
15546:Ch. 24.
15514:3027812
15383:(ed.),
15333:1967772
15299:2972885
15263:2301359
15218:3619777
15161:2323624
14346:
14329:
14319:
14305:
14029:to the
14001:of the
13956:
13940:
13927:
13911:
13898:
13882:
13161:gives
13159:+ 1 = 0
12415:
12399:
11729:+ 1 = 0
11608:
11586:
11521:
11501:
11488:
11468:
11455:
11435:
11134:
11117:
11100:
11083:
9505:of the
9196:
9182:
9176:
9162:
9156:
9142:
9135:of the
9037:√
9025:√
8960:
8943:
8929:
8912:
8837:
8822:
8812:
8800:
8412:
8396:
8171:, when
6837:cosines
6609:
6596:√
6591:
6436:
6423:√
6418:
6314:
6302:
6012:
5997:
5964:
5949:
3879:or, if
934:factors
641:Persian
461:Liu Hui
429:History
383:quartic
323:of the
235:algebra
15945:degree
15841:
15833:
15801:
15793:
15759:
15739:
15731:
15699:
15691:
15657:
15638:
15630:
15622:
15590:
15582:
15574:
15539:
15512:
15476:
15438:
15408:
15360:
15331:
15297:
15261:
15224:
15216:
15208:
15159:
15074:
15049:
14983:
14958:
14910:
14867:
14813:604533
14811:
14747:
14717:
14653:
14625:
14597:
14536:
14509:
14476:
14453:
14430:
14401:
14264:+ 2000
14161:, the
14095:matrix
11491:, and
9548:81 = 9
9395:where
8845:= 1, 2
8735:arccos
8605:arccos
8500:of an
8496:, and
8243:> 0
8238:where
8077:
8018:arsinh
7943:
7935:
7897:
7820:arcosh
7725:, as
7592:arccos
7168:
6959:arccos
6584:where
6200:" or "
6163:
6158:
6130:
6125:
5861:(Both
5371:
5366:
5338:
5333:
5158:
5153:
5125:
5120:
4498:where
2749:roots.
2641:, and
1454:with
1000:, and
980:. The
976:being
495:conics
342:, and
15839:S2CID
15799:S2CID
15737:S2CID
15697:S2CID
15636:S2CID
15628:JSTOR
15588:S2CID
15580:JSTOR
15510:JSTOR
15329:JSTOR
15295:JSTOR
15259:JSTOR
15222:S2CID
15214:JSTOR
15186:(PDF)
15168:(PDF)
15157:JSTOR
15137:(PDF)
15047:S2CID
14956:S2CID
14934:(PDF)
14908:S2CID
14809:JSTOR
14593:–91.
14256:+ 200
14215:Notes
14143:of a
14133:of a
14033:of a
13836:with
11709:be a
11310:Let
9223:over
8727:Here
8414:) = 0
8112:When
6894:are
6594:–1 +
6421:–1 ±
6411:by a
5709:field
5593:field
5186:field
4271:field
3066:is a
2712:roots
1930:roots
1890:of a
1872:roots
1656:roots
1272:Let
968:with
519:) = 2
509:Graph
499:roots
416:with
414:field
321:roots
46:roots
15934:and
15831:ISSN
15791:ISSN
15757:ISBN
15729:ISSN
15689:ISSN
15655:ISBN
15620:ISSN
15572:ISSN
15537:ISBN
15474:ISBN
15436:ISBN
15406:ISBN
15358:ISBN
15206:ISSN
15072:ISBN
14981:ISBN
14865:ISBN
14745:ISBN
14715:ISBN
14651:ISBN
14623:ISBN
14595:ISBN
14534:ISBN
14507:ISBN
14474:ISBN
14451:ISBN
14428:ISBN
14399:ISBN
14303:) =
14260:= 20
14104:The
14089:The
14018:area
14016:The
13999:foci
13979:and
13930:and
13859:and
13678:and
13150:and
13136:and
13088:and
13029:and
12978:and
12887:and
12791:and
12760:and
12646:and
12558:and
12485:and
12467:and
12449:and
12427:and
12090:and
11836:and
11720:and
11611:for
11582:and
11408:and
10862:<
10807:>
9895:and
9610:and
9574:and
9536:) =
9417:and
9133:foci
9027:tan
8967:and
8840:for
8449:and
8207:= 16
8137:= −3
8117:= ±3
8083:>
7997:sinh
7949:<
7929:>
7799:cosh
7119:<
6839:and
6146:and
5888:and
5721:Let
5587:and
5354:and
5230:with
5141:and
5065:<
4945:and
4705:See
4626:and
4525:and
4375:are
4371:and
4308:and
3772:and
3329:≠ 0
3324:and
3267:= 0
3131:<
3104:and
3041:and
3006:>
2727:<
2692:>
2609:real
2309:is
2182:is
1934:real
1886:The
1824:for
1654:The
988:and
972:and
736:and
701:= 20
697:+ 10
682:Flos
667:root
665:the
637:+ 35
629:+ 12
481:and
455:, a
373:and
237:, a
43:real
15943:By
15876:on
15823:doi
15783:doi
15721:doi
15681:doi
15612:doi
15564:doi
15502:doi
15321:doi
15287:doi
15251:doi
15198:doi
15149:doi
15039:doi
14948:doi
14900:doi
14861:220
14801:doi
14797:110
14503:306
14391:doi
14227:In
14172:In
14157:In
14116:or
13938:= −
13880:= −
13867:= −
13844:= 0
13148:= 1
12382:By
11718:= 1
11696:= 0
11681:= 0
11627:= 0
11421:of
11070:= 0
10570:As
10383:):
10363:or
10162:so
9604:= 0
9544:− 1
9540:− 3
9215:of
9029:ORH
8932:so
8686:for
8584:cos
8539:= 0
8200:= 2
8176:= 3
8163:1/3
8149:1/3
8107:≠ 0
8102:If
7719:≠ 0
7571:cos
7457:cos
7384:cos
7363:cos
7233:cos
7221:cos
7200:cos
7164:= 0
7148:.
7139:= 0
6937:cos
6249:= 0
6209:= 0
6190:any
6186:any
5994:is
5946:is
5711:of
5616:≠ 0
5595:of
5468:≠ 0
5461:= 0
5454:= 0
5197:/ 3
5104:In
5028:If
4273:of
3254:If
2716:If
2681:If
1905:If
1008:).
820:in
693:+ 2
661:to
633:= 6
597:= 0
590:≠ 0
233:In
53:= 0
21:two
16150::
15886:–
15865:,
15859:,
15837:,
15829:,
15817:,
15815:92
15813:,
15797:,
15789:,
15777:,
15775:92
15773:,
15751:,
15735:,
15727:,
15719:,
15715:,
15713:93
15711:,
15695:,
15687:,
15675:,
15673:91
15671:,
15634:,
15626:,
15618:,
15604:78
15602:,
15586:,
15578:,
15570:,
15556:81
15554:,
15531:,
15508:,
15496:,
15468:,
15464:,
15453:^
15445:,
15415:,
15356:,
15352:,
15327:,
15317:19
15315:,
15293:,
15283:25
15281:,
15257:,
15247:42
15245:,
15220:,
15212:,
15204:,
15194:77
15192:,
15188:,
15155:.
15145:95
15143:.
15139:.
15045:.
15035:92
15033:.
15006:.
14989:.
14954:.
14944:90
14942:.
14936:.
14919:^
14906:,
14896:24
14894:,
14863:.
14839:,
14833:,
14829:,
14807:,
14795:,
14777:,
14771:,
14767:,
14681:,
14591:87
14560:^
14505:.
14442:^
14412:^
14397:,
14385:,
14348:–
14327:=
14299:+
14243:,
14237:,
14233:,
14176:,
13909:=
13901:,
13854:=
13846:,
13807:27
13625:12
13155:+
12721:0.
12386:,
12081:,
11827:,
11725:+
11698:.
11692:+
11690:pt
11688:+
11677:+
11675:cx
11673:+
11671:bx
11669:+
11667:ax
11633:.
11623:=
11584:−
11549:27
11499:−
11466:−
11458:,
11433:−
11399:,
11354:27
11296:0.
11288:27
11220::
11196:0.
11177:27
11115:–
11111:=
11103:.
11081:–
11077:=
11066:+
11064:pt
11062:+
10984:.
10849:27
10794:27
10743:27
10672:27
10543:27
10450:27
10340:27
10268:27
10212:0.
10204:27
9781:0.
9741:0.
9600:+
9598:pt
9596:+
9562:.
9550:.
9496:.
9408:,
9112:±h
9089:hi
9087:±
9058:DA
9054:=
9050:BE
9046:=
9041:RH
9034:=
9022:=
9014:OM
9011:=
9006:,
9003:hi
9001:±
8941:+
8937:=
8910:−
8906:=
8535:+
8533:pt
8531:+
8526:,
8492:,
8394:−
8386:+
8254:=
8245:,
8233:=
8226:+
8202:,
8178:.
8086:0.
7916:27
7695:2.
7437:0.
7254:0.
7173:=
7160:+
7158:pt
7156:+
7141:.
7106:27
7064:2.
6797:0.
6612:.
6598:–3
6589:=
6425:–3
6006:12
6000:−1
5952:−1
5827:27
5702:,
5698:,
5694:,
5618:)
5607:A
5569:0.
5553:27
5450:=
5294:27
5052:27
4684:27
4610:27
4419:27
4289:.
3581:.
3293:27
3263:=
3242:0.
3226:27
3134:0.
3112:–
3096:–
3079:–
3058:–
3009:0.
2783:,
2776:,
2633:,
2626:,
2577:27
2549:27
2399:27
2317:18
2215:27
1920:,
1913:,
1902:.
1878:.
1862:.
1620:27
1602:27
1393:.
1215:.
1199:/
787:=
785:mx
783:+
773:=
771:mx
769:+
727:=
725:mx
723:+
599:.
586:,
576:=
574:qx
572:+
570:px
568:+
425:.
389:.)
369:,
338:,
334:,
15924:e
15917:t
15910:v
15880:.
15825::
15785::
15723::
15683::
15614::
15566::
15504::
15498:5
15323::
15289::
15253::
15200::
15151::
15080:.
15053:.
15041::
15016:.
14962:.
14950::
14902::
14873:.
14803::
14723:.
14631:.
14603:.
14542:.
14515:.
14482:.
14436:.
14393::
14361:.
14355:w
14350:w
14342:w
14340:3
14337:/
14333:p
14325:t
14316:3
14313:/
14309:p
14301:w
14297:t
14295:(
14293:w
14288:w
14262:x
14258:x
14254:x
14086:.
14056:7
14052:/
14045:2
13952:a
13948:/
13944:d
13936:3
13933:e
13923:a
13919:/
13915:c
13907:2
13904:e
13894:a
13890:/
13886:b
13878:1
13875:e
13869:q
13865:3
13862:e
13856:p
13852:2
13849:e
13842:1
13839:e
13820:,
13815:3
13811:e
13804:+
13799:2
13795:e
13789:1
13785:e
13781:9
13773:3
13768:1
13764:e
13760:2
13757:=
13750:S
13743:,
13738:2
13734:e
13730:3
13722:2
13717:1
13713:e
13709:=
13702:P
13680:S
13676:P
13658:.
13653:2
13649:x
13643:1
13639:x
13633:0
13629:x
13622:+
13619:)
13614:2
13609:0
13605:x
13599:2
13595:x
13591:+
13586:2
13581:2
13577:x
13571:1
13567:x
13563:+
13558:2
13553:1
13549:x
13543:0
13539:x
13535:+
13530:0
13526:x
13520:2
13515:2
13511:x
13507:+
13502:2
13498:x
13492:2
13487:1
13483:x
13479:+
13474:1
13470:x
13464:2
13459:0
13455:x
13451:(
13448:3
13442:)
13437:3
13432:2
13428:x
13424:+
13419:3
13414:1
13410:x
13406:+
13401:3
13396:0
13392:x
13388:(
13385:2
13382:=
13377:3
13372:2
13368:s
13364:+
13359:3
13354:1
13350:s
13346:=
13339:S
13332:,
13329:)
13324:0
13320:x
13314:2
13310:x
13306:+
13301:2
13297:x
13291:1
13287:x
13283:+
13278:1
13274:x
13268:0
13264:x
13260:(
13252:2
13247:2
13243:x
13239:+
13234:2
13229:1
13225:x
13221:+
13216:2
13211:0
13207:x
13203:=
13198:2
13194:s
13188:1
13184:s
13180:=
13173:P
13157:ξ
13153:ξ
13146:ξ
13138:P
13134:S
13115:.
13112:v
13109:3
13106:=
13101:2
13097:s
13076:u
13073:3
13070:=
13065:1
13061:s
13040:,
13037:v
13027:u
13013:.
13010:p
13004:3
13001:1
12992:=
12989:v
12986:u
12966:v
12963:+
12960:u
12957:=
12952:0
12948:x
12927:,
12924:p
12921:3
12915:=
12910:2
12906:s
12900:1
12896:s
12875:)
12870:2
12866:s
12862:+
12857:1
12853:s
12849:(
12843:3
12840:1
12834:=
12829:0
12825:x
12802:.
12799:v
12789:u
12773:2
12769:s
12746:1
12742:s
12718:=
12713:3
12709:P
12705:+
12702:z
12699:S
12691:2
12687:z
12664:3
12659:2
12655:s
12632:3
12627:1
12623:s
12600:3
12595:2
12591:s
12587:+
12582:3
12577:1
12573:s
12569:=
12566:S
12546:,
12541:2
12537:s
12531:1
12527:s
12523:=
12520:P
12506:i
12504:s
12499:i
12497:s
12491:2
12488:s
12482:1
12479:s
12473:2
12470:s
12464:1
12461:s
12455:2
12452:x
12446:1
12443:x
12433:2
12430:s
12424:1
12421:s
12411:a
12407:/
12403:b
12397:−
12392:0
12389:s
12364:.
12361:)
12356:2
12352:s
12346:2
12338:+
12333:1
12329:s
12322:+
12317:0
12313:s
12309:(
12303:3
12300:1
12294:=
12285:2
12281:x
12273:,
12270:)
12265:2
12261:s
12254:+
12249:1
12245:s
12239:2
12231:+
12226:0
12222:s
12218:(
12212:3
12209:1
12203:=
12194:1
12190:x
12182:,
12179:)
12174:2
12170:s
12166:+
12161:1
12157:s
12153:+
12148:0
12144:s
12140:(
12134:3
12131:1
12125:=
12116:0
12112:x
12096:2
12093:s
12087:1
12084:s
12078:0
12075:s
12056:,
12051:2
12047:x
12040:+
12035:1
12031:x
12025:2
12017:+
12012:0
12008:x
12004:=
11995:2
11991:s
11983:,
11978:2
11974:x
11968:2
11960:+
11955:1
11951:x
11944:+
11939:0
11935:x
11931:=
11922:1
11918:s
11910:,
11905:2
11901:x
11897:+
11892:1
11888:x
11884:+
11879:0
11875:x
11871:=
11862:0
11858:s
11842:2
11839:x
11833:1
11830:x
11824:0
11821:x
11805:,
11800:3
11796:/
11789:i
11786:2
11782:e
11778:=
11773:2
11767:3
11762:i
11756:1
11747:=
11727:ξ
11723:ξ
11716:ξ
11707:ξ
11694:q
11686:t
11679:d
11631:0
11625:q
11621:p
11614:i
11603:i
11599:w
11597:3
11594:/
11590:p
11578:i
11574:w
11558:.
11552:W
11543:3
11539:p
11517:3
11514:w
11512:3
11509:/
11505:p
11497:3
11494:w
11484:2
11481:w
11479:3
11476:/
11472:p
11464:2
11461:w
11451:1
11448:w
11446:3
11443:/
11439:p
11431:1
11428:w
11423:W
11414:3
11411:w
11405:2
11402:w
11396:1
11393:w
11374:4
11369:2
11365:q
11359:+
11349:3
11345:p
11332:2
11329:q
11321:=
11318:W
11293:=
11283:3
11279:p
11270:)
11265:3
11261:w
11257:(
11254:q
11251:+
11246:2
11242:)
11236:3
11232:w
11228:(
11218:w
11213:w
11193:=
11185:3
11181:w
11171:3
11167:p
11158:q
11155:+
11150:3
11146:w
11130:w
11128:3
11125:/
11121:p
11113:w
11109:t
11096:w
11094:3
11091:/
11087:p
11079:w
11075:t
11068:q
11060:t
11027:.
11022:2
11016:3
11005:1
10968:.
10962:u
10959:3
10954:p
10945:=
10942:v
10922:,
10917:3
10914:p
10906:=
10903:v
10900:u
10868:,
10865:0
10857:2
10853:q
10846:+
10841:3
10837:p
10833:4
10813:,
10810:0
10802:2
10798:q
10791:+
10786:3
10782:p
10778:4
10752:3
10738:3
10734:p
10728:+
10723:4
10718:2
10714:q
10701:2
10698:q
10687:+
10681:3
10667:3
10663:p
10657:+
10652:4
10647:2
10643:q
10635:+
10630:2
10627:q
10616:=
10613:t
10593:,
10590:t
10587:=
10584:v
10581:+
10578:u
10558:.
10552:3
10538:3
10534:p
10528:+
10523:4
10518:2
10514:q
10501:2
10498:q
10487:=
10484:v
10465:.
10459:3
10445:3
10441:p
10435:+
10430:4
10425:2
10421:q
10413:+
10408:2
10405:q
10394:=
10391:u
10371:v
10361:u
10347:.
10335:3
10331:p
10325:+
10320:4
10315:2
10311:q
10298:2
10295:q
10262:3
10258:p
10254:4
10248:+
10243:2
10239:q
10235:=
10209:=
10199:3
10195:p
10186:x
10183:q
10180:+
10175:2
10171:x
10144:3
10140:)
10136:v
10133:u
10130:(
10127:+
10124:x
10121:)
10116:3
10112:v
10108:+
10103:3
10099:u
10095:(
10087:2
10083:x
10079:=
10067:3
10063:v
10057:3
10053:u
10049:+
10046:x
10043:)
10038:3
10034:v
10030:+
10025:3
10021:u
10017:(
10009:2
10005:x
10001:=
9991:)
9986:3
9982:v
9975:x
9972:(
9969:)
9964:3
9960:u
9953:x
9950:(
9947:=
9940:0
9913:,
9908:3
9904:v
9892:u
9871:.
9866:3
9863:p
9855:=
9848:v
9845:u
9838:q
9832:=
9823:3
9819:v
9815:+
9810:3
9806:u
9778:=
9775:p
9772:+
9769:v
9766:u
9763:3
9738:=
9735:q
9732:+
9729:)
9726:v
9723:+
9720:u
9717:(
9714:)
9711:p
9708:+
9705:v
9702:u
9699:3
9696:(
9693:+
9688:3
9684:v
9680:+
9675:3
9671:u
9650:t
9647:=
9644:v
9641:+
9638:u
9618:v
9608:u
9602:q
9594:t
9542:x
9538:x
9534:x
9532:(
9530:p
9524:3
9521:A
9515:3
9512:S
9502:Q
9494:K
9489:3
9486:A
9480:3
9477:A
9423:3
9420:r
9414:2
9411:r
9405:1
9402:r
9397:a
9383:,
9380:)
9375:3
9371:r
9362:2
9358:r
9354:(
9351:)
9346:3
9342:r
9333:1
9329:r
9325:(
9322:)
9317:2
9313:r
9304:1
9300:r
9296:(
9291:2
9287:a
9283:=
9266:Δ
9258:3
9255:A
9249:3
9246:S
9239:(
9237:K
9233:K
9225:K
9213:K
9193:3
9190:/
9186:π
9173:3
9170:/
9166:π
9153:3
9150:/
9146:π
9105:x
9097:g
9085:g
9080:x
9078:(
9062:.
9020:h
9009:g
8999:g
8977:t
8973:y
8969:x
8965:t
8956:a
8954:3
8951:/
8947:b
8939:x
8935:t
8925:a
8923:3
8920:/
8916:b
8908:t
8904:x
8899:t
8895:)
8892:1
8889:(
8868:3
8865:p
8855:2
8843:k
8834:3
8831:/
8827:π
8825:2
8819:k
8817:−
8809:3
8806:/
8803:1
8784:)
8777:p
8773:3
8760:p
8757:2
8752:q
8749:3
8742:(
8713:.
8709:2
8706:,
8703:1
8700:,
8697:0
8694:=
8691:k
8680:)
8674:3
8667:2
8661:k
8654:)
8647:p
8643:3
8630:p
8627:2
8622:q
8619:3
8612:(
8600:3
8597:1
8591:(
8577:3
8574:p
8564:2
8561:=
8556:k
8552:t
8537:q
8529:t
8524:)
8521:2
8518:(
8512:.
8506:x
8498:C
8494:B
8490:A
8486:x
8482:)
8479:1
8476:(
8419:y
8408:m
8404:/
8400:n
8392:x
8390:(
8388:x
8384:y
8378:y
8363:.
8359:)
8355:x
8345:2
8341:m
8337:n
8331:(
8327:x
8324:=
8317:2
8313:m
8307:4
8303:x
8289:m
8287:/
8285:x
8277:x
8273:x
8269:x
8260:m
8258:/
8256:x
8252:y
8241:n
8235:n
8231:x
8228:m
8224:x
8215:x
8211:2
8205:n
8198:m
8174:p
8169:)
8167:q
8165:(
8160:S
8155:)
8153:q
8151:(
8146:C
8135:p
8125:0
8122:t
8115:p
8105:p
8080:p
8067:]
8062:)
8055:p
8052:3
8043:p
8040:2
8035:q
8032:3
8025:(
8013:3
8010:1
8004:[
7991:3
7988:p
7982:2
7976:=
7967:0
7963:t
7955:,
7952:0
7946:p
7932:0
7924:2
7920:q
7913:+
7908:3
7904:p
7900:4
7887:]
7882:)
7875:p
7871:3
7858:p
7855:2
7849:|
7845:q
7841:|
7837:3
7827:(
7815:3
7812:1
7806:[
7792:3
7789:p
7777:q
7772:|
7768:q
7764:|
7757:2
7751:=
7742:0
7738:t
7717:p
7692:,
7689:1
7686:,
7683:0
7680:=
7677:k
7667:]
7661:3
7657:k
7651:2
7641:)
7634:p
7630:3
7617:p
7614:2
7609:q
7606:3
7599:(
7587:3
7584:1
7578:[
7563:3
7560:p
7549:2
7546:=
7541:k
7537:t
7516:,
7509:p
7505:3
7492:p
7489:2
7484:q
7481:3
7475:=
7472:)
7466:3
7463:(
7434:=
7428:p
7424:3
7410:p
7407:2
7402:q
7399:3
7381:3
7367:3
7359:4
7339:.
7334:4
7329:3
7325:u
7302:,
7294:3
7291:p
7280:2
7277:=
7274:u
7251:=
7248:)
7242:3
7239:(
7218:3
7204:3
7196:4
7186:u
7182:.
7179:θ
7175:u
7171:t
7162:q
7154:t
7137:p
7122:0
7114:2
7110:q
7103:+
7098:3
7094:p
7090:4
7061:,
7058:1
7055:,
7052:0
7049:=
7046:k
7036:]
7029:3
7025:k
7019:2
7009:)
7001:p
6997:3
6984:p
6981:2
6976:q
6973:3
6966:(
6954:3
6951:1
6944:[
6929:3
6926:p
6915:2
6912:=
6907:k
6903:t
6882:0
6879:=
6876:q
6873:+
6870:t
6867:p
6864:+
6859:3
6855:t
6794:=
6789:1
6781:=
6776:0
6750:.
6745:3
6741:)
6734:a
6731:3
6727:b
6722:+
6719:x
6716:(
6713:a
6692:,
6686:a
6683:3
6678:b
6652:,
6649:0
6646:=
6641:1
6633:=
6628:0
6606:2
6603:/
6587:ξ
6571:,
6567:}
6564:2
6561:,
6558:1
6555:,
6552:0
6549:{
6543:k
6539:,
6535:)
6528:C
6523:k
6512:0
6502:+
6499:C
6494:k
6486:+
6483:b
6479:(
6472:a
6469:3
6465:1
6457:=
6452:k
6448:x
6433:2
6430:/
6409:C
6405:C
6388:.
6383:)
6377:C
6372:0
6362:+
6359:C
6356:+
6353:b
6349:(
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6327:=
6324:x
6311:0
6308:/
6305:0
6288:,
6285:0
6282:=
6277:1
6269:=
6264:0
6247:C
6232:0
6229:=
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6207:C
6202:–
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6169:3
6101:,
6095:3
6090:2
6082:3
6077:0
6069:4
6061:2
6056:1
6041:1
6029:=
6026:C
6008:a
6003:/
5980:0
5960:a
5958:8
5955:/
5932:1
5901:1
5874:0
5843:.
5840:d
5835:2
5831:a
5824:+
5821:c
5818:b
5815:a
5812:9
5804:3
5800:b
5796:2
5793:=
5784:1
5772:,
5769:c
5766:a
5763:3
5755:2
5751:b
5747:=
5738:0
5704:d
5700:c
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5673:0
5670:=
5667:d
5664:+
5661:x
5658:c
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5650:2
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5639:+
5634:3
5630:x
5626:a
5614:a
5589:q
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5566:=
5561:2
5557:q
5550:+
5545:3
5541:p
5537:4
5513:.
5510:q
5487:,
5484:0
5481:=
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5466:C
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5452:q
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5430:.
5425:2
5419:3
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5309:.
5303:3
5289:3
5285:p
5279:+
5274:4
5269:2
5265:q
5257:+
5252:2
5249:q
5238:=
5235:C
5222:C
5219:3
5215:p
5207:C
5195:p
5193:–
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5071:,
5068:0
5060:2
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5008:3
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4982:+
4976:3
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4904:+
4898:3
4892:1
4888:u
4880:1
4855:.
4850:2
4844:3
4839:i
4833:1
4824:=
4819:2
4814:1
4806:=
4801:2
4776:,
4771:2
4765:3
4760:i
4757:+
4754:1
4745:=
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4691:.
4679:3
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4664:4
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4448:u
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4410:p
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4399:4
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4390:q
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4352:=
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4340:p
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4332:3
4328:t
4249:.
4244:2
4240:)
4234:2
4230:x
4223:x
4220:(
4217:)
4212:1
4208:x
4201:x
4198:(
4195:a
4192:=
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4186:+
4183:x
4180:c
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4172:2
4168:x
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4161:+
4156:3
4152:x
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4128:.
4122:)
4119:c
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4101:b
4097:(
4094:a
4087:3
4083:b
4076:d
4071:2
4067:a
4063:9
4057:c
4054:b
4051:a
4048:4
4042:=
4037:1
4033:x
4012:,
4006:)
4003:c
4000:a
3997:3
3989:2
3985:b
3981:(
3978:2
3973:c
3970:b
3964:d
3961:a
3958:9
3952:=
3947:3
3943:x
3939:=
3934:2
3930:x
3909:,
3906:c
3903:a
3900:3
3892:2
3888:b
3863:3
3858:)
3851:a
3848:3
3844:b
3839:+
3836:x
3832:(
3827:a
3824:=
3821:d
3818:+
3815:x
3812:c
3809:+
3804:2
3800:x
3796:b
3793:+
3788:3
3784:x
3780:a
3760:,
3754:a
3751:3
3747:b
3739:=
3734:3
3730:x
3726:=
3721:2
3717:x
3713:=
3708:1
3704:x
3683:,
3680:c
3677:a
3674:3
3671:=
3666:2
3662:b
3637:d
3634:+
3631:x
3628:c
3625:+
3620:2
3616:x
3612:b
3609:+
3604:3
3600:x
3596:a
3559:.
3554:2
3549:)
3542:p
3539:2
3534:q
3531:3
3525:+
3522:t
3518:(
3512:)
3506:p
3502:q
3499:3
3490:t
3486:(
3482:=
3479:q
3476:+
3473:t
3470:p
3467:+
3462:3
3458:t
3435:.
3429:p
3426:2
3421:q
3418:3
3409:=
3404:3
3400:t
3396:=
3391:2
3387:t
3362:p
3358:q
3355:3
3349:=
3344:1
3340:t
3327:p
3312:,
3309:0
3306:=
3301:2
3297:q
3290:+
3285:3
3281:p
3277:4
3265:q
3261:p
3256:p
3239:=
3234:2
3230:q
3223:+
3218:3
3214:p
3210:4
3190:q
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3178:+
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3114:r
3110:1
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3101:2
3098:r
3094:1
3091:r
3086:)
3084:3
3081:r
3077:2
3074:r
3072:(
3063:3
3060:r
3056:2
3053:r
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3044:r
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3035:r
3029:1
3026:r
2979:2
2975:)
2969:3
2965:r
2956:2
2952:r
2948:(
2943:2
2939:)
2933:3
2929:r
2920:1
2916:r
2912:(
2907:2
2903:)
2897:2
2893:r
2884:1
2880:r
2876:(
2871:4
2867:a
2863:=
2840:d
2837:+
2834:x
2831:c
2828:+
2823:2
2819:x
2815:b
2812:+
2807:3
2803:x
2799:a
2788:3
2785:r
2781:2
2778:r
2774:1
2771:r
2755:r
2733:,
2730:0
2698:,
2695:0
2643:a
2638:3
2635:r
2631:2
2628:r
2624:1
2621:r
2593:.
2585:2
2581:a
2570:2
2566:)
2562:d
2557:2
2553:a
2546:+
2543:c
2540:b
2537:a
2534:9
2526:3
2522:b
2518:2
2515:(
2507:3
2503:)
2499:c
2496:a
2493:3
2485:2
2481:b
2477:(
2474:4
2448:4
2444:a
2423:.
2418:2
2414:d
2408:2
2404:a
2391:3
2387:c
2383:a
2379:4
2371:2
2367:c
2361:2
2357:b
2353:+
2350:d
2345:3
2341:b
2336:4
2330:d
2327:c
2324:b
2321:a
2297:d
2294:+
2291:x
2288:c
2285:+
2280:2
2276:x
2272:b
2269:+
2264:3
2260:x
2256:a
2234:.
2230:)
2224:2
2220:q
2212:+
2207:3
2203:p
2198:4
2194:(
2170:q
2167:+
2164:t
2161:p
2158:+
2153:3
2149:t
2126:.
2121:2
2117:)
2111:3
2107:r
2098:2
2094:r
2090:(
2085:2
2081:)
2075:3
2071:r
2062:1
2058:r
2054:(
2049:2
2045:)
2039:2
2035:r
2026:1
2022:r
2018:(
2013:4
2009:a
1988:,
1985:d
1982:+
1979:x
1976:c
1973:+
1968:2
1964:x
1960:b
1957:+
1952:3
1948:x
1944:a
1925:3
1922:r
1918:2
1915:r
1911:1
1908:r
1850:3
1847:,
1844:2
1841:,
1838:1
1835:=
1832:i
1812:,
1806:a
1803:3
1799:b
1789:i
1785:t
1781:=
1776:i
1772:x
1749:3
1745:t
1741:,
1736:2
1732:t
1728:,
1723:1
1719:t
1696:3
1692:x
1688:,
1683:2
1679:x
1675:,
1670:1
1666:x
1636:.
1628:3
1624:a
1615:d
1610:2
1606:a
1599:+
1596:c
1593:b
1590:a
1587:9
1579:3
1575:b
1571:2
1561:=
1558:q
1546:2
1542:a
1538:3
1531:2
1527:b
1520:c
1517:a
1514:3
1504:=
1501:p
1491:a
1488:3
1484:b
1479:+
1476:x
1469:=
1466:t
1442:,
1439:0
1436:=
1433:q
1430:+
1427:t
1424:p
1421:+
1416:3
1412:t
1398:a
1390:t
1385:t
1368:a
1365:3
1361:b
1353:t
1350:=
1347:x
1327:0
1324:=
1321:d
1318:+
1315:x
1312:c
1309:+
1304:2
1300:x
1296:b
1293:+
1288:3
1284:x
1280:a
1253:q
1250:+
1247:t
1244:p
1241:+
1236:3
1232:t
1201:q
1197:p
1178:3
1174:q
1167:2
1163:p
1159:a
1156:+
1153:q
1150:p
1147:b
1144:+
1139:2
1135:q
1131:c
1125:+
1122:x
1115:2
1111:q
1106:p
1103:a
1100:+
1097:q
1094:b
1088:+
1083:2
1079:x
1073:q
1070:a
1043:,
1038:q
1035:p
1030:=
1025:1
1021:x
1006:d
1002:p
998:a
994:q
990:p
986:q
974:p
970:q
956:p
950:x
947:q
912:,
909:0
906:=
903:d
900:+
897:x
894:c
891:+
886:2
882:x
878:b
875:+
870:3
866:x
862:a
789:n
781:x
775:n
767:x
738:n
734:m
729:n
721:x
699:x
695:x
691:x
651:(
635:x
631:x
627:x
595:q
588:q
584:p
578:N
566:x
541:x
537:x
533:x
529:x
525:x
521:x
517:x
515:(
513:f
408:.
344:d
340:c
336:b
332:a
314:a
300:0
297:=
294:d
291:+
288:x
285:c
282:+
277:2
273:x
269:b
266:+
261:3
257:x
253:a
213:)
210:4
207:+
204:x
201:(
198:)
195:1
192:+
189:x
186:(
183:)
180:2
174:x
171:(
166:4
163:1
158:=
147:)
143:8
137:x
134:6
126:2
122:x
118:3
115:+
110:3
106:x
101:(
95:4
92:1
87:=
80:)
77:x
74:(
71:f
51:y
34:.
27:.
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