Cubic function From Wikipedia, the free encyclopedia

This article discusses cubic equations in one variable. For a discussion of cubic equations in two variables, see elliptic curve.

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 of a cubic function.

History Cubic equations were known to the ancient Indians and ancient Greeks since the 5th century BC, and even earlier to the ancient Babylonians who were able to solve certain cubic equations, and ancient Egyptians, who dealt with the problem of doubling the cube, and attempted to solve it using compass and straightedge constructions.[1] Hippocrates, Menaechmus and Archimedes are believed to have come close to solving this problem using intersecting conic sections,[1] 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 were the roots are 0, 1 or 2.[2]

Graph of a cubic function with its 3 roots, i.e. where the curve

crosses the x-axis (y = 0). It has 2 critical points.

Contents 1 History 2 Roots of a cubic function

2.1 The nature of the roots 2.2 General formula of roots 2.3 Monic formula of roots 2.4 Cardano's method

2.4.1 Summary 2.4.2 Alternate method

2.5 Lagrange resolvents 2.5.1 Description 2.5.2 Procedure

2.6 Factorization 2.7 Root-finding formula 2.8 Solution in terms of Chebyshev radicals

2.8.1 The case of a cubic equation with real coefficients3 Derivative 4 Bipartite cubics 5 See also 6 Notes 7 References 8 External links

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, that it cannot be solved using earlier compass and straightedge constructions, and found a geometric solution which could be used to get a numerical answer by

Two-dimensional graph of a cubic, the polynomial ƒ(x) = 2x3 − 3x2 − 3x + 2.

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.[3][4]

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.[5] He understood the importance of the discriminant of the cubic equation and used an early version of Cardano's formula[6] to find algebraic solutions to certain types of cubic equations.[7]

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.[8]

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 [9].

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.

Roots of a cubic function

The nature of the roots

Every cubic equation 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: [10]

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.

See also: multiplicity of a root of a polynomial

General formula of roots

For the general cubic equation ax3 + bx2 + cx + d = 0, the general formulas for the roots, in terms of the coefficients, are as follows:[11]

Monic formula of roots

For a monic polynomial (dividing by the leading coefficient), renaming the coefficients as x3 + ax2 + bx + c = 0, this reduces to

Introducing the following notation

simplifies the above equations to

In this form, they can be identified as the inverse discrete Fourier transform of the triple:

which is the perspective taken in the method of Lagrange resolvents.

Since there are 3 possible values for each cube root, and must be taken so

that they satisfy αβ = k.

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.

We first divide the standard equation by the leading coefficient to arrive at an equation of the form

The substitution eliminates the quadratic term, giving the so-called depressed cubic

where

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

which, combined with (3) (the first parenthesis vanishes, then multiply by u3 and substitute uv) gives

This can be seen as a quadratic equation in u3. When we solve this equation, we find that

and thus

Since t = v + u, t = x + a/3, and v = −p/3u, we find

Note that there are six possibilities in computing u with (4), since there are two possibilities for the square root ( ),

and three for the cubic root (the principal root and the principal root multiplied by ). The sign of the square root however does not affect the resulting t (a simple calculation shows that −p/3u = v), although care must be taken in three special cases to avoid divisions by zero:

First, if p = q = 0, then we have the triple real root

Second, if p = 0 and q ≠ 0, then

Third, if p ≠ 0 and q = 0 then

in which case the three roots are

where

Summary

In summary, for the cubic equation

the solutions for x are given by

where

The expression above for u can generate up to three values (there are three cubic roots related by a factor which is one of the two non-real cubic roots of one, and two square roots of any sign ; but these 6 expressions can generate only 3 pairs). This also applies to the final solutions for x.

Alternate method

An alternate method to obtain the same results is as follows.

We know that

Since u and v must satisfy

,

it can be shown that

Writing out the three cube roots we get

Remembering t = u + v we get only three possible values for t because only three combinations of u and v are

possible if is to remain valid as it must — so

and x is obtained from

The above methods do apply if p and q are complex. This solution avoids the addition of an inverted cubic radical in the solution, and also resolves the ambiguity of signs for the square roots in the first solution given above.

Summary

To simplify the expressions above, it is customary to define this resolution in several steps by defining intermediate variables. Let

and

Then the discriminant of the quadratic equation of or is

,

Let's also define a constant that represents a generator for the three cubic roots of unity:

Then the solutions for x = t - A can be simply defined for k in 0, 1, 2 in :

where are the two possible values for or .

In the case p and q are both real, the following cases can be distinguished, according to the sign of the discriminant.

1. If D is strictly positive then there are one real root and two non-real, conjugate roots. 2. If D is zero then there is one real root (a triple root) or two real roots (a single root and a double root.) 3. If D is strictly negative then there are three real roots (Casus irreducibilis).

Lagrange resolvents

Description

Lagrange resolvents, introduced by Joseph Louis Lagrange in his paper Réflexions sur la résolution algébrique des équations, reduced the solution of a cubic equation to the solution of an auxiliary quadratic polynomial, the "resolving equation" or resolvent quadratic of the original equation,[12][13] by a change of variables from the roots x0,x1,x2 to the Lagrange resolvents r0,r1,r2. In modern terms, the resolvents are the discrete Fourier transform of the original roots.

In the modern language of Galois theory, Lagrange exploited the fact that the symmetric group S3 has the cyclic

group of order three C3 as a normal subgroup, thus allowing one to decompose the permutation group of the roots as

– the C3 corresponds to the discrete Fourier transform (of order 3), while the

C2 = S2 corresponds to the quadratic resolving polynomial.

In the same way, the symmetric group of order four S4 has a Klein four-group as normal subgroup, with quotient a symmetric group of order three S3, which allows one to solve a quartic in terms of a cubic resolving polynomial.

However, this method does not work for polynomials of degree five or greater, as the resolving polynomial has higher degree than the original polynomial (for a quintic the resolving polynomial has degree 24[12]). This is explained by the Abel–Ruffini theorem, which proves that such polynomials cannot be solved by radicals.

Procedure

Suppose that r0, r1 and r2 are the roots of equation (1), and define , so that ζ is a primitive third root of unity. 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

We already know the value s0 = −a, so we only need to seek values for the other two.

The si are not symmetric in the roots – s0 is invariant, while the two non-trivial cyclic permutations of the roots send

s1 to ζs1 and s2 to ζ2s2, or s1 to ζ2s1 and s2 to ζs2 (depending on which permutation), while transposing r1 and r2 switches s1 and s2 – other transpositions switch these roots and multiplying them by a power of ζ.

Thus, if one takes the cubes, the factors of ζ become factors of ζ3 = 1, so every cyclic permutation leaves the cubes invariant, and a transposition of two roots exchanges s1

3 and s23 – in other words, S3 (permutation of the roots) acts

as a permutation group on specifically by the sign permutation exchanging the last two, meaning that the action factors through the sign map corresponding to the cyclic group of order 3 being a normal subgroup.

Hence the polynomial

is invariant under permutations of the roots, and so has coefficients expressible in terms of (1). Using calculations involving symmetric functions or alternatively field extensions, we can calculate (5) to be

The roots of this quadratic equation are

where D is the discriminant. Taking cube roots give us s1 and s2, from which we can recover the roots ri of (1).

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.

Root-finding formula

The formula for finding the roots of a cubic function, based on Cardano's method, is fairly complicated. Therefore, it is common to use the rational root test or a numerical solution instead.

If we have

with and , let

and

we define the discriminant:

There are two distinct cases:

in which case there are one real root and two non-real roots that are conjugate. We define:

and

in which case we have 3 real roots. We express the complex quantity in polar form:

and we define:

and

In both cases, the solutions are

Solution in terms of Chebyshev radicals

If we have a cubic equation which is already in depressed form, we may write it as . Substituting we obtain or equivalently

From this we obtain solutions to our original equation in terms of the Chebyshev cube root as

If now we start from a general equation

and reduce it to the depressed form under the substitution x = t − a/3, we have and , leading to

This gives us the solutions to (1) as

The case of a cubic equation with real coefficients

Suppose the coefficients of (1) are real. If s is the quantity q/r from the section on real roots, then s = t2; hence 0 < s < 4 is equivalent to −2 < t < 2, and in this case we have a polynomial with three distinct real roots, expressed in terms of a real function of a real variable, quite unlike the situation when using cube roots. If s > 4 then either t > 2 and

is the sole real root, or t < −2 and is the sole real root. If s < 0 then the reduction to Chebyshev polynomial form has given a t which is a pure imaginary number; in this case is the sole real root. We are now evaluating a real root by means of a function of a purely imaginary argument; however we can avoid this by using the function

which is a real function of a real variable with no singularities along the real axis. If a polynomial can be reduced to

the form x3 + 3x − t with real t, this is a convenient way to solve for its roots.

Derivative Through the quadratic formula the roots of the derivative

,

are given by

and provide the critical points where the slope of the cubic equation 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 is strictly monotonic.

Bipartite cubics The graph of

where 0<a<b , is called a bipartite cubic. This is from the theory of elliptic curves.

One can graph a bipartite cubic on a graphing device by graphing the function

corresponding to the upper half of the bipartite cubic. It is defined on

See also Linear equation Quadratic equation Quartic equation Quintic equation Polynomial Newton's method Spline (mathematics) Viete

Notes

1. ^ a b Guilbeau (1930).

"The Egyptians considered the solution impossible, but the Greeks came nearer to a solution."

2. ^ The works of Archimedes, translation by T. L. Heath 3. ^ J. J. O'Connor and E. F. Robertson (1999), Omar Khayyam, MacTutor History of Mathematics archive.

"Khayyam himself seems to have been the first to conceive a general theory of cubic equations."

4. ^ Guilbeau (1930). "Omar Al Hay of Chorassan, about 1079 AD did most to elevate to a method the

solution of the algebraic equations by intersecting conics."

5. ^ O'Connor, John J.; Robertson, Edmund F., "Sharaf al-Din al-Muzaffar al-Tusi", MacTutor History of Mathematics archive, University of St Andrews, http://www-history.mcs.st-andrews.ac.uk/Biographies/Al-Tusi_Sharaf.html.

6. ^ Rashed, Roshdi; Armstrong, Angela (1994), The Development of Arabic Mathematics, Springer, pp. 342–3, ISBN 0792325656

7. ^ 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

8. ^ "The life and numbers of Fibonacci" [1], Plus Magazine 9. ^ Katz, Victor. A History of Mathematics. pp. 220. Boston: Addison Wesley, 2004.

10. ^ Irving, Ronald S. (2004), Integers, polynomials, and rings, Springer-Verlag New York, Inc., ISBN 0-387-40397-3, http://books.google.com/books?id=B4k6ltaxm5YC, Chapter 10 ex 10.14.4 and 10.17.4, p. 154-156

11. ^ http://hk.knowledge.yahoo.com/question/question?qid=7007111502216 12. ^ a b Prasolov, Viktor; Solovyev, Yuri (1997), Elliptic functions and elliptic integrals, AMS Bookstore, ISBN 978 0

82180587 9, http://books.google.com/books?id=fcp9IiZd3tQC, §6.2, p. 134 13. ^ Kline, Morris (1990), Mathematical Thought from Ancient to Modern Times, Oxford University Press US, ISBN 978 0

19506136 9, http://books.google.com/books?id=aO-v3gvY-I8C, Algebra in the Eighteenth Century: The Theory of Equations

References W. S. Anglin; & J. Lambek (1995). "Mathematics in the Renaissance", in The heritage of Thales, Ch. 24. Springers. Lucye Guilbeau (1930). "The History of the Solution of the Cubic Equation", Mathematics News Letter 5 (4), p. 8-12. R.W.D. Nickalls (1993). A new approach to solving the cubic: Cardan's solution revealed, The Mathematical Gazette, 77:354–359.

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 on 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

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