Cubic equation

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

Graph of a cubic polynomial:
y = x³/4 + 3x²/4 − 3x/2 − 2
= (1/4)(x + 4)(x + 1)(x − 2)

In mathematics, a cubic equation is a polynomial equation in which the highest occurring power of the unknown is the third power. The standard form of a cubic equation is

$ax^{3}+bx^{2}+cx+d=0\,$.

Usually, the coefficients a,..., d are real numbers. However, most of the theory is also valid if they belong to a field of characteristic other than two or three. We will always assume that a is non-zero (otherwise it is a quadratic equation).

Solving a cubic equation amounts to finding the roots of a cubic function.

History

Cubic equations were discovered by the ancient civilisations more or less independently. In the Greek world, it was known by the fifth century BCE with problems such as doubling the cube. The Jaina mathematicians in ancient India is sometimes credited with the discovery sometime between 400 BCE and 200 CE.

According to some books on the history of mathematics, Menaechmus and Archimedes invented the method of solving cubic equations by intersecting conic sections, though historians such as Reviel Netz disputed whether the Greeks were thinking about cubic equations or just problems that can lead to cubic equations. The Persian mathematician Omar Khayyám (1048–1131) re-discovered the method of constructing solutions of cubic equations by intersecting a parabola with a circle. He showed how this geometric solution could be used to get a numerical answer by consulting trigonometric tables.

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 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. Tartaglia did so only on the condition that Cardano would never reveal it. A few years later, Cardano learned about Ferro's prior work and broke the promise by publishing Tartaglia's method in his book Ars Magna (1545) with credit given to Tartaglia. This led to another competition between Tartaglia and Cardano, for which the latter did not show up but was represented by his student Lodovico Ferrari (1522-1565). Ferrari did better than Tartaglia in the competition, and Tartaglia lost both his prestige and income.

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.

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,

$\Delta =4b^{3}d-b^{2}c^{2}+4ac^{3}-18abcd+27a^{2}d^{2}.\,$

The following cases need to be considered.

• If Δ < 0, then the equation has three distinct real roots.
• If Δ > 0, then the equation has one real root and a pair of complex conjugate roots.
• If Δ = 0, then (at least) two roots coincide. It may be that the equation has a double real root and another distinct single real root; alternatively, all three roots coincide yielding a triple real root. A possible way to decide between these subcases is to compute the resultant of the cubic and its second derivative: a triple root exists if and only if this resultant vanishes.

See also: multiplicity of a root of a polynomial

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 given equation by α₃ to arrive at an equation of the form

$x^{3}+ax^{2}+bx+c=0.\qquad (1)$

The substitution x = t - a/3 eliminates the quadratic term; in fact, we get the equation

$t^{3}+pt+q=0,\quad {\mbox{where }}p=b-{\frac {a^{2}}{3}}\quad {\mbox{and}}\quad q=c+{\frac {2a^{3}-9ab}{27}}.\qquad (2)$

This is called the depressed cubic.

Suppose that we can find numbers u and v such that

$u^{3}-v^{3}=q\quad {\mbox{and}}\quad uv={\frac {p}{3}}.\qquad (3)$

A solution to our equation is then given by

$t=v-u,\,$

as can be checked by directly substituting this value for t in (2), as a consequence of the third order binomial identity

$(v-u)^{3}+3uv(v-u)+(u^{3}-v^{3})=0\ .$

The system (3) can be solved by solving the second equation for v, which gives

$v={\frac {p}{3u}}.$

Substituting this in the first equation in (3) yields

$u^{3}-{\frac {p^{3}}{27u^{3}}}=q.$

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

$u={\sqrt[{3}]{{q \over 2}\pm {\sqrt {{q^{2} \over 4}+{p^{3} \over 27}}}}}.\qquad (4)$

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

$x={\frac {p}{3u}}-u-{a \over 3}.$

Note that there are six possibilities in computing u with (4), since there are two solutions to the square root ($\pm$), and three complex solutions to the cubic root (the principal root and the principal root multiplied by $-1/2\pm i{\sqrt {3}}/2$). However, which sign of the square root is chosen does not affect the final resulting x, although care must be taken in two special cases to avoid divisions by zero. First, if p = 0, then one should choose the sign of the square root that gives a nonzero value for u, i.e. $u={\sqrt[{3}]{q}}$. Second, if p = q = 0, then we have the triple real root x = −a/3.

Lagrange resolvents

The symmetric group S₃ of order three has the cyclic group of order three as a normal subgroup, which suggests making use of the discrete Fourier transform of the roots, an idea due to Lagrange. Suppose that r₀, r₁ and r₂ are the roots of equation (1), and define $\zeta =(-1+i{\sqrt {3}})/2$, so that ζ is a primitive third root of unity. We now set

$s_{0}=r_{0}+r_{1}+r_{2},\,$

$s_{1}=r_{0}+\zeta r_{1}+\zeta ^{2}r_{2},\,$

$s_{2}=r_{0}+\zeta ^{2}r_{1}+\zeta r_{2}.\,$

The roots may then be recovered from the three s_i by inverting the above linear transformation, giving

$r_{0}=(s_{0}+s_{1}+s_{2})/3,\,$

$r_{1}=(s_{0}+\zeta ^{2}s_{1}+\zeta s_{2})/3,\,$

$r_{2}=(s_{0}+\zeta s_{1}+\zeta ^{2}s_{2})/3.\,$

We already know the value s₀ = −a, so we only need to seek values for the other two. However, if we take the cubes, a cyclic permutation leaves the cubes invariant, and a transposition of two roots exchanges s₁³ and s₂³, hence the polynomial

$(z-s_{1}^{3})(z-s_{2}^{3})\qquad (5)$

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

${z}^{2}+\left(-9\,ba+2\,{a}^{3}+27\,c\right)z+\left({a}^{2}-3\,b\right)^{3}.$

The roots of this quadratic equation are

${\frac {9}{2}}\,ab-{a}^{3}-{\frac {27}{2}}\,c\pm {\frac {3}{2}}\,{\sqrt {3\Delta }},$

where Δ is the discriminant defined above. Taking cube roots give us s₁ and s₂, from which we can recover the roots r_i of (1).

Factorization

If r is any root of (1), then we may factor using r to obtain

$(x-r)(x^{2}+(a+r)x+b+ar+r^{2})=x^{3}+ax^{2}+bx+c.$

Hence if we know one root we can find the other two by solving a quadratic equation, giving

${\frac {1}{2}}\left(-a-r\pm {\sqrt {-3r^{2}-2ar+a^{2}-4b}}\right)$

for the other two roots.

Solution in terms of Chebyshev radicals

If we have a cubic equation which is already in depressed form, we may write it as $\,x^{3}-3px-q=0$. Substituting $x={\sqrt {p}}z$ we obtain $z^{3}-3z-p^{-{\frac {3}{2}}}q=0$ or equivalently

$z^{3}-3z=p^{-{\frac {3}{2}}}q\ .$

From this we obtain solutions to our original equation in terms of the Chebyshev cube root $C_{1 \over 3}$ as

$r_{0}={\sqrt {p}}\,C_{1 \over 3}(p^{-{\frac {3}{2}}}q),\,$

$r_{1}=-{\sqrt {p}}\,C_{1 \over 3}(-p^{-{\frac {3}{2}}}q),\,$

$r_{2}=-r_{0}-r_{1}\ .$

If now we start from a general equation

$x^{3}+ax^{2}+bx+c=0\qquad (1)$

and reduce it to the depressed form under the substitution x = t − a/3, we have $\,p=(a^{2}-3b)/9$ and $\,q=-(2a^{3}-9ab+27c)/27$, leading to

$t_{a;b;c}=p^{-{\frac {3}{2}}}q=-{\frac {2a^{3}-9ab+27c}{(a^{2}-3b)^{3/2}}}.$

This gives us the solutions to (1) as

$r_{0}={\sqrt {p}}\,C_{1 \over 3}(t_{a;b;c})-{a \over 3},\,$

$r_{1}=-{\sqrt {p}}\,C_{1 \over 3}(-t_{a;b;c})-{a \over 3},\,$

$r_{2}=-r_{0}-r_{1}-a\ .$

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 $C_{1 \over 3}(t)$ is the sole real root, or t < −2 and $-C_{1 \over 3}(-t)$ 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 $iC_{1 \over 3}(-it)-iC_{1 \over 3}(it)$ 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

$S_{1 \over 3}(t)=iC_{1 \over 3}(-it)-iC_{1 \over 3}(it)=2\operatorname {sinh} \left(\operatorname {arcsinh} \left({t \over 2}\right)/3\right),\,$

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 $x^{3}+3x-t$ with real t, this is a convenient way to solve for its roots.

See also

• Cubic function
• Linear equation
• Quadratic equation
• Quartic equation
• Quintic equation

External links

• 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

References

• W. S. Anglin; & J. Lambek (1995). Mathematics in the Renaissance. In The heritage of Thales, Ch. 24. Springers.
• R.W.D. Nickalls (1993). A new approach to solving the cubic: Cardan's solution revealed, The Mathematical Gazette, 77:354–359.