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{{about|cubic equations in one variable|cubic equations in ''two'' variables|elliptic curve}} |
{{about|cubic equations in one variable|cubic equations in ''two'' variables|elliptic curve}} |
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[[Image:Polynomialdeg3.svg|thumb|right|200px|Graph of a cubic function with |
[[Image:Polynomialdeg3.svg|thumb|right|200px|Graph of a cubic function with 3 [[real number|real]] [[root of a function|roots]] (where the curve crosses the horizontal axis—where ''y'' = 0). It has 2 [[critical point (mathematics)|critical points]]. Here the function is <math>y=x^3/3+3 x^2 /4-2 x/3-2</math>.]] |
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In [[mathematics]], a '''cubic function''' is a [[function (mathematics)|function]] of the form |
In [[mathematics]], a '''cubic function''' is a [[function (mathematics)|function]] of the form |
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Revision as of 15:20, 28 October 2011
In mathematics, a cubic function is a function of the form
where a is nonzero; or in other words, a polynomial of degree three. The derivative of a cubic function is a quadratic function. The integral of a cubic function is a quartic function.
Setting ƒ(x) = 0 and assuming a ≠ 0 produces a cubic equation of the form:
Usually, the coefficients a, b,c, d are real numbers. However, most of the theory is also valid if they belong to a field of characteristic other than 2 or 3. Solving a cubic equation amounts to finding the roots or zeros of a cubic function. There are two ways to solve a cubic equation. One way is to express the roots as formulas involving simple functions like square and cube roots. This is the approach which is described in this page. Another way is to use a numerical approximation of the roots in the field of the real or complex numbers. This may be obtained by any root-finding algorithm, like Newton's method.
History
Cubic equations were known to ancient Greek mathematician Diophantus;[1] even earlier to ancient Babylonians who were able to solve certain cubic equations;[2] and also to the ancient Egyptians. Doubling the cube is the simplest and oldest studied cubic equation, and one which the ancient Egyptians considered to be impossible.[3] Hippocrates 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 compass and straightedge construction,[4] a task which is now known to be impossible. Hippocrates, Menaechmus and Archimedes are believed to have come close to solving the problem of doubling the cube using intersecting conic sections,[4] 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 T. L. Heath, who translated all Archimedes' works, disagree, putting forward evidence that Archimedes really solved cubic equations using intersections of two cones, but also discussed the conditions where the roots are 0, 1 or 2.[5]
In the 7th century, the Tang dynasty astronomer mathematician Wang Xiaotong in his mathematical treatise titled Jigu Suanjing systematically established and solved 25 cubic equations of the form , 23 of them with , and two of them with .[6]
In the 11th century, the Persian poet-mathematician, Omar Khayyám (1048–1131), made significant progress in the theory of cubic equations. In an early paper he wrote regarding cubic equations, 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 which could be used to get a numerical answer by consulting trigonometric tables. In his later work, the Treatise on Demonstration of Problems of Algebra, he wrote a complete classification of cubic equations with general geometric solutions found by means of intersecting conic sections.[7][8]
In the 12th century, the Indian mathematician Bhaskara II attempted cubic equations without general success. However, he gave one example of a cubic equation:[9]
In the 12th century, another Persian mathematician, Sharaf al-Dīn al-Tūsī (1135–1213), wrote the Al-Mu'adalat (Treatise on Equations), 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 "Ruffini-Horner method" to numerically approximate the root of a cubic equation. He also developed the concepts of a derivative function and the maxima and minima of curves in order to solve cubic equations which may not have positive solutions.[10] He understood the importance of the discriminant of the cubic equation to find algebraic solutions to certain types of cubic equations.[11]
Leonardo de Pisa, also known as Fibonacci (1170–1250), was able to find the positive solution to the cubic equation x3+2x2+10x = 20, using the Babylonian numerals. He gave the result as 1,22,7,42,33,4,40 which is equivalent to: 1+22/60+7/602+42/603+33/604+4/605+40/606.[12]
In the early 16th century, the Italian mathematician Scipione del Ferro (1465–1526) found a method for solving a class of cubic equations, namely those of the form x3 + mx = n. In fact, all cubic equations can be reduced to this form if we allow m and n to be negative, but negative numbers 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 Fiore about it.
In 1530, Niccolò Tartaglia (1500–1557) received two problems in cubic equations from Zuanne da Coi and announced that he could solve them. He was soon challenged by Fiore, 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 money. Tartaglia received questions in the form x3 + mx = n, for which he had worked out a general method. Fiore received questions in the form x3 + mx2 = n, which proved to be too difficult for him to solve, and Tartaglia won the contest.
Later, Tartaglia was persuaded by Gerolamo Cardano (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 reveal a book about cubics, that he would give Tartaglia time to publish. Some years later, Cardano learned about Ferro's prior work and published Ferro's method in his book Ars Magna in 1545, meaning Cardano gave Tartaglia 6 years to publish his results (with credit given to Tartaglia for an independent solution). Cardano's promise with Tartaglia stated that he 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 by Tartaglia, which Cardano denied. The challenge was eventually accepted by Cardano's student Lodovico Ferrari (1522–1565). Ferrari did better than Tartaglia in the competition, and Tartaglia lost both his prestige and income.[13]
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 complex numbers in Ars Magna, but he did not really understand it. Rafael Bombelli studied this issue in detail and is therefore often considered as the discoverer of complex numbers.
Derivative
Through the quadratic formula the roots of the derivative f′(x) = 3ax2 + 2bx + c are given by
and provide the critical points where the slope of the cubic function is zero. If b2 − 3ac > 0, then the cubic function has a local maximum and a local minimum. If b2 − 3ac = 0, then the cubic's inflection point is the only critical point. If b2 − 3ac < 0, then there are no critical points. In the cases where b2 − 3ac ≤ 0, the cubic function is strictly monotonic.
Roots of a cubic function
The general cubic equation has the form
with
This section describes how the roots of such an equation may be computed. The coefficients a, b, c, d are generally assumed to be real numbers, but most of the results apply when they belong to any field of characteristic not 2 or 3.
The nature of the roots
Every cubic equation (1) with real coefficients has at least one solution x among the real numbers; this is a consequence of the intermediate value theorem. We can distinguish several possible cases using the discriminant,
The following cases need to be considered: [14]
- If Δ > 0, then the equation has three distinct real roots.
- If Δ = 0, then the equation has a multiple root and all its roots are real.
- If Δ < 0, then the equation has one real root and two nonreal complex conjugate roots.
General formula of roots
For the general cubic equation (1) with real coefficients, the general formula for the roots, in terms of the coefficients, is as follows if , i.e. if there are two non real roots:
However, this formula is applicable only when the operand of the square root is non-negative and a,b,c,d are real coefficients. When this operand is real and positive, the cubic roots are real and well defined. In the other case, the square root is not real and one has to choose, once for all a determination for it, for example the one with positive imaginary part. For extracting the cubic roots we have also to choose a determination for the cubic roots, and this gives nine possible values for the first root of an equation which has only three roots.
A correct solution may be obtained by remarking that the proof of above formula shows that the product of the two cubic roots is rational. This gives the following formula in which or stand for any determination of the square or cubic root, if
If and b2 − 3ac = 0, the sign of Q has to be chosen for having .
If and , the three roots are equal:
If and , above expression for the roots is correct but misleading, hiding the fact that no radical is needed to represent the roots. In fact, in this case, there is a double root
-
- and a simple root
The next sections describe how these formulas may be obtained.
Reduction to a monic trinomial
Dividing Equation (1) by and substituting by (Tschirnhaus transformation) we get the equation
where
Any formula for the roots of Equation (2) may be transformed in a formula for the roots of Equation (1) by substituting and by the above values and using the relation .
Therefore, only Equation (2) is considered in the following.
Cardano's method
The solutions can be found with the following method due to Scipione del Ferro and Tartaglia, published by Gerolamo Cardano in 1545.[15]
We first apply preceding reduction, giving the so-called depressed cubic
We introduce two variables u and v linked by the condition
and substitute this in the depressed cubic (2), giving
- .
At this point Cardano imposed a second condition for the variables u and v
- .
As the first parenthesis vanishes in (3), we get and . Thus and are the two roots of the equation
At this point, Cardano, who did not know complex numbers, supposed that the roots of this equation were real, that is that
Solving this equation and using the fact that and may be exchanged, we find
- and .
As these expressions are real, their cubic roots are well defined and, like Cardano, we get
The two complex roots are obtained by considering the complex cubic roots; the fact is real implies that they are obtained by multiplying one of the above cubic roots by and the other by .
If is not necessarily positive, we have to choose a cubic root of . As there is no direct way to choose the corresponding cubic root of , one has to use the relation , which gives
and
Note that the sign of the square root does not affect the resulting , because changing it amounts to exchanging and . We have chosen the minus sign to have when and , in order to avoid a division by zero. With this choice, the above expression for works always, except when , where the second term becomes 0/0. In this case there is a triple root .
Note also that in several cases the solutions are expressed with less square or cubic roots
- If then we have the triple real root
- If and then
- and the three roots are the three cubic roots of .
- If and then
- in which case the three roots are
- where
- Finally if , there is a double root and a simple root which may be expressed rationally in term of , but this expression may not be immediately deduced from the general expression of the roots:
To pass from these roots of in Equation (2) to the general formulas for roots of in Equation (1), subtract and replace and by their expressions in terms of .
Lagrange's method
In his paper Réflexions sur la résolution algébrique des équations (Thoughts on the algebraic solving of equations), Joseph Louis Lagrange introduced a new method to solve the equations of low degree.
This method works well for cubic and quartic equations, but Lagrange did not succeed in applying it to a quintic equation, because it implies to solve a resolving polynomial of degree at least six.[16][17][18] This is explained by the Abel–Ruffini theorem, which proves that such polynomials cannot be solved by radicals. Nevertheless the modern methods for solving solvable quintic equations are mainly based on Lagrange's method.[18]
In the case of cubic equations, Lagrange's method gives the same solution as Cardano's, but avoids its magic aspect (Why did Cardano choose these auxiliary variables?). Moreover it may also be applied directly to general cubic equation (1) without using the reduction to the trinomial equation (2). Nevertheless the computation is much easier on this reduced equation.
Suppose that x0, x1 and x2 are the roots of equation (1) or (2), and define , so that ζ is a primitive third root of unity which satisfies the relation . We now set
This is the discrete Fourier transform of the roots: observe that while the coefficients of the polynomial are symmetric in the roots, in this formula an order has been chosen on the roots, so these are not symmetric in the roots. The roots may then be recovered from the three si by inverting the above linear transformation via the inverse discrete Fourier transform, giving
The polynomial is an elementary symmetric polynomial and is thus equal to in case of Equation (1) and to zero in case of Equation (2), so we only need to seek values for the other two.
The polynomials and are not symmetric functions of the roots: is invariant, while the two non-trivial cyclic permutations of the roots send to and to , or to and to (depending on which permutation), while transposing and switches and ; other transpositions switch these roots and multiply them by a power of
Thus, , and are left invariant by the cyclic permutations of the roots, which multiply them by . Also and are left invariant by the transposition of and which exchanges and . As the permutation group of the roots is generated by these permutations, it follows that and are symmetric functions of the roots and may thus be written as polynomials in the elementary symmetric polynomials and thus as rational functions of the coefficients of the equation. Let and these expressions, which will be explicitly computed below.
We have got that and are the two roots of the quadratic equation
Thus the resolution of the equation may be finished exactly as described for Cardano's method, with and in place of and .
Computation of A and B
Setting , and , the elementary symmetric polynomials, we have, using that :
The expression for is the same with and exchanged. Thus, using we get
and a straightforward computation gives
Similarly we have
When solving Equation (1) we have
- , and
With Equation (2), we have , and and thus:
- and .
Note that with Equation (2), we have and , when in Cardano's method we have set and Thus we have, up to the exchange of and :
- and .
In other words, in this case, Cardano's 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.
Trigonometric (and hyperbolic) method
When a cubic equation has three real roots, the formulas expressing these roots in term of radicals involve complex numbers. A representation of these roots in term of cosine and arccosine allows to avoid complex numbers. The formulas which follow are true in general (except when p=0), but involve complex cosine and arccosine when there is only one real root.
Starting from Equation (2), , let us set The idea is to choose for identifying Equation (2) with the identity
In fact, choosing and dividing Equation (2) by we get
Combining with above identity, we get
and thus the roots are[19]
This formula is totally real if and the argument of the arccosine is between -1 and 1. The last condition is equivalent with which implies also . Thus the above formula for the roots is totally real if and only if the three roots are real.
Denoting by the above value of t0 and using the inequality for a real number u such that the three roots may also be expressed as
If the three roots are real, we have
All these formulas may be straightforwardly transformed into formulas for the roots of the general cubic equation (1), using the back substitution described in Section Reduction to a monic trinomial.
When there is only one real root (and p≠0), it may be similarly represented, using hyperbolic functions, as[20]
If p≠0 and the inequalities on the right are not satisfied the formulas remain valid but involve complex quantities.
When , above values of are sometimes named Chebyshev cube root.[21] More precisely the values involving cosines and hyperbolic cosines define, when , the same analytic function denoted , which is the proper Chebyshev cube root. The value involving hyperbolic sines is similarly denoted when .
Factorization
If r is any root of (1), then we may factor using r to obtain
Hence if we know one root we can find the other two by solving a quadratic equation, giving
for the other two roots.
See also
- Casus irreducibilis
- Linear equation
- Quadratic equation
- Quartic equation
- Quintic equation
- Polynomial
- Newton's method
- Spline (mathematics)
- Viete
Notes
- ^ Van de Waerden, Geometry and Algebra of Ancient Civilizations, chapter 4, Zurich 1983 ISBN 0387121595
- ^ British Museum BM 85200
- ^ Guilbeau (1930, p. 8) states, "The Egyptians considered the solution impossible, but the Greeks came nearer to a solution."
- ^ a b Guilbeau (1930, pp. 8–9)
- ^ The works of Archimedes, translation by T. L. Heath
- ^ Mikami, Yoshio (1974) [1913], "Chapter 8 Wang Hsiao-Tung and Cubic Equations", The Development of Mathematics in China and Japan (2nd ed.), New York: Chelsea Publishing Co., pp. 53–56, ISBN 978-0828401494
- ^ J. J. O'Connor and E. F. Robertson (1999), Omar Khayyam, MacTutor History of Mathematics archive, states, "Khayyam himself seems to have been the first to conceive a general theory of cubic equations."
- ^ Guilbeau (1930, 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."
- ^ Datta and Singh, History of Hindu Mathematics, p76,Equation of Higher Degree; Bharattya Kala Prakashan, Delhi, India 2004 ISBN 81-86050-86-8
- ^ O'Connor, John J.; Robertson, Edmund F., "Sharaf al-Din al-Muzaffar al-Tusi", MacTutor History of Mathematics Archive, University of St Andrews
- ^ J. L. Berggren (1990), "Innovation and Tradition in Sharaf al-Din al-Tusi's Muadalat", Journal of the American Oriental Society 110 (2): 304–9
- ^ "The life and numbers of Fibonacci" [1], Plus Magazine
- ^ Katz, Victor. A History of Mathematics. pp. 220. Boston: Addison Wesley, 2004.
- ^ Irving, Ronald S. (2004), Integers, polynomials, and rings, Springer-Verlag New York, Inc., ISBN 0-387-40397-3, Chapter 10 ex 10.14.4 and 10.17.4, p. 154-156
- ^ Jacobson (2009), p. 210.
- ^ Prasolov, Viktor; Solovyev, Yuri (1997), Elliptic functions and elliptic integrals, AMS Bookstore, ISBN 978 0 82180587 9, §6.2, p. 134
- ^ Kline, Morris (1990), Mathematical Thought from Ancient to Modern Times, Oxford University Press US, ISBN 978 0 19506136 9, Algebra in the Eighteenth Century: The Theory of Equations
- ^ a b Daniel Lazard, "Solving quintics in radicals", in Olav Arnfinn Laudal, Ragni Piene, The Legacy of Niels Henrik Abel, pp. 207–225, Berlin, 2004,. ISBN 3-5404-3826-2
- ^ Shelbey, Samuel (1975), CRC Standard Mathematical Tables, CRC Press, ISBN 0 87819 622 6
- ^ These are Formulas (80) and (83) of Weisstein, Eric W. 'Cubic Formula'. From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/CubicFormula.html, rewritten for having a coherent notation.
- ^ Abramowitz, Milton; Stegun, Irene A., eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover (1965), chap. 22 p. 773
References
- Anglin, W. S.; Lambek, Joachim (1995), "Mathematics in the Renaissance", The Heritage of Thales, Springers, pp. 125–131, ISBN 978-0387945446 Ch. 24.
- Guilbeau, Lucye (1930), "The History of the Solution of the Cubic Equation", Mathematics News Letter, 5 (4): 8–12, doi:10.2307/3027812, JSTOR 3027812
- Jacobson, Nathan (2009), Basic algebra, vol. 1 (2nd ed.), Dover, ISBN 978-0-486-47189-1
- Nickalls, R. W. D. (November 1993), "A new approach to solving the cubic: Cardan's solution revealed" (PDF), The Mathematical Gazette, 77 (480): 354–359, doi:10.2307/3619777, ISSN 0025-5572, JSTOR 3619777
- Henriquez, G., "The graphical interpretation of the complex roots of cubic equations," American Mathematical Monthly 42, June–July 1935, 383-384.
- Dunnett, R., "Newton-Raphson and the cubic," Mathematical Gazette 78, November 1994, 347-348.
- Dence, T., "Cubics, chaos and Newton's method," Mathematical Gazette 81, November 1997, 403-408.
- Holmes, G. C., "The use of hyperbolic cosines in solving cubic polynomials," Mathematical Gazette 86, November 2002, 473-477.
- Nickalls, R. W. D., "Viète, Descartes and the cubic equation," Mathematical Gazette 90, July 2006, 203-208.
- Mitchell, D. W., "Solving cubics by solving triangles," Mathematical Gazette 91, November 2007, 514-516.
- Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007), "Section 5.6 Quadratic and Cubic Equations", Numerical Recipes: The Art of Scientific Computing (3rd ed.), New York: Cambridge University Press, ISBN 978-0-521-88068-8
- Zucker, I. J., "The cubic equation—A new look at the irreducible case," Mathematical Gazette 92, July 2008, 264-268.
- Rechtschaffen, E., "Real roots of cubics: Explicit formula for quasi-solutions," Mathematical Gazette 92, July 2008, 268-276.
- Mitchell, D. W., "Powers of as roots of cubics," Mathematical Gazette 93, November 2009.
External links
- Solving a Cubic by means of Moebius transforms
- Interesting derivation of trigonometric cubic solution with 3 real roots
- Calculator for solving Cubics (also solves Quartics and Quadratics)
- Tartaglia's work (and poetry) on the solution of the Cubic Equation at Convergence
- Cubic Equation Solver.
- Quadratic, cubic and quartic equations on MacTutor archive.
- "Cubic Formula". PlanetMath.
- Cardano solution calculator as java applet at some local site. Only takes natural coefficients.
- Graphic explorer for cubic functions With interactive animation, slider controls for coefficients
- On Solution of Cubic Equations at Holistic Numerical Methods Institute
- Dave Auckly, Solving the quartic with a pencil American Math Monthly 114:1 (2007) 29—39
- "Cubic Equation" by Eric W. Weisstein, The Wolfram Demonstrations Project, 2007.
- The affine equivalence of cubic polynomials at Dynamic Geometry Sketches