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goniometrical

Cubic function

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

f(x)=ax^3+bx^2+cx+d,,

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.

If you set f(x)=0, you get a cubic equation of the form:

ax^3+bx^2+cx+d=0 ,

where

ane 0 ,

(if a = 0, then the equation becomes a quadratic equation).
(if both a and b = 0, then the equation becomes a linear equation).

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 two or three.

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 Egyptians, who dealt with the problem of doubling the cube, and attempted to solve it using compass and straightedge constructions. Hippocrates, Menaechmus and Archimedes are believed to have come close to solving this problem using intersecting conic sections, though historians such as Reviel Netz dispute whether the Greeks were thinking about cubic equations or just problems that can lead to cubic equations.

In the 11th century, the Persian poet-mathematician, Omar Khayyám (10481131), 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 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.

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

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 .

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,
Delta = -4b^3d + b^2c^2 - 4ac^3 + 18abcd - 27a^2d^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 standard equation by the leading coefficient 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, quadmbox{where } p = b - frac{a^2}3 quadmbox{and}quad q = c + frac{2a^3-9ab}{27}. qquad (2)
This is called the depressed cubic. A simple and elegant way of solving the depressed cubic is due to Thomas Harriot (1560 – 1621): substituting t=y-{pover 3y} into it and multiplying both sides by y^3 yields, after much cancellation, y^6+q y^3-{p^3over 27}=0. Described below is the original, somewhat long-winded method of Cardano and Tartaglia, which still dominates the textbooks today.

Suppose that we can find numbers u and v such that

-u^3-v^3 = q quadmbox{and}quad -uv = frac{p}{3}. quad (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 into the first equation in (3) yields
- u^3 + frac{p^3}{27u^3} = q.
Moving all to the q side and multiplying by 27u3 yields
27u^6 + 27qu^3 - p^3 = 0,.
This can be seen as a quadratic equation for u3. If we solve this equation, we find that
u^{3}=-{qover 2}pm sqrt{{q^{2}over 4}+{p^{3}over 27}}
u=sqrt[3]{-{qover 2}pm sqrt{{q^{2}over 4}+{p^{3}over 27}}}. quad (4)
Since t = v + u, t = x + a/3, and v = −p/3u, we find
x=-frac{p}{3u}+u-{aover 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 tfrac{-1}{2} pm itfrac{sqrt{3}}{2}. However, the sign of the square root (plus or minus) 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 negative square root so that u does not equal zero, i.e., u = -sqrt[3]{q}. Second, if p = q = 0, then we have the triple real root x = −a/3.

In summary, for the cubic equation

x^3 + ax^2 + bx +c = 0
the solutions for x are given by
x=-frac{p}{3u}+u-{aover 3}
where
p = b - frac{a^2}3
q = c + frac{2a^3-9ab}{27}
u=sqrt[3]{-{qover 2}pm sqrt{{q^{2}over 4}+{p^{3}over 27}}}.

Although this method is simple and elegant, it fails for the case of three real roots, e.g. when:

D < 0, D = left ({q over 2} right )^2 + left ({p over 3} right )^3
For this case a different method (e.g. goniometrical) has to be used.

Actually this method can work for the case where D < 0 and all other cases if we use all three cube roots of u and v above whether real or complex. This is important historically because finding solutions this way helped lead to the acceptance of complex numbers. Unfortunately things then were a bit more complicated.

text{We know that }!!~!!text{ }!!~!!text{ }!!~!!text{ }!!~!!text{ }u^{3}=-frac{q}{2}+sqrt{frac{q^{2}}{4}+frac{p^{3}}{27}}~~~~or~~~~-frac{q}{2}-sqrt{frac{q^{2}}{4}+frac{p^{3}}{27}} .

text{But since u and v must satisfy }!!~!!text{ }!!~!!text{ }!!~!!text{ }!!~!!text{ }-u^{3}-v^{3}=q~~~~and~~~~-uv=frac{p}{3}

then one can show that if

text{ }~~~~~~u^{3}=-frac{q}{2}+sqrt{frac{q^{2}}{4}+frac{p^{3}}{27}}~~~~then~~~~v^{3}=-frac{q}{2}-sqrt{frac{q^{2}}{4}+frac{p^{3}}{27}}

or vice versa it doesn't matter which so we will chose this way.

Writing out the three cube roots we get

u=left{ begin{align} & sqrt[3]{-frac{q}{2}+sqrt{frac{q^{2}}{4}+frac{p^{3}}{27}}} & left(-frac{1}{2}+ifrac{sqrt{3}}{2} right)sqrt[3]{-frac{q}{2}+sqrt{frac{q^{2}}{4}+frac{p^{3}}{27}}} & left(-frac{1}{2}-ifrac{sqrt{3}}{2} right)sqrt[3]{-frac{q}{2}+sqrt{frac{q^{2}}{4}+frac{p^{3}}{27}}} end{align} right.~~~and~~~v=left{ begin{align} & sqrt[3]{-frac{q}{2}-sqrt{frac{q^{2}}{4}+frac{p^{3}}{27}}} & left(-frac{1}{2}+ifrac{sqrt{3}}{2} right)sqrt[3]{-frac{q}{2}-sqrt{frac{q^{2}}{4}+frac{p^{3}}{27}}} & left(-frac{1}{2}-ifrac{sqrt{3}}{2} right)sqrt[3]{-frac{q}{2}-sqrt{frac{q^{2}}{4}+frac{p^{3}}{27}}} end{align} right.

Remembering ~t=u+v~ we get only three possible values for t because only three combinations of u and v are possible if -uv=frac{p}{3} is to remain valid as it must - so

t=left{ begin{align} & sqrt[3]{-frac{q}{2}+sqrt{frac{q^{2}}{4}+frac{p^{3}}{27}}}+sqrt[3]{-frac{q}{2}-sqrt{frac{q^{2}}{4}+frac{p^{3}}{27}}} & left(-frac{1}{2}+ifrac{sqrt{3}}{2} right)sqrt[3]{-frac{q}{2}+sqrt{frac{q^{2}}{4}+frac{p^{3}}{27}}}+left(-frac{1}{2}-ifrac{sqrt{3}}{2} right)sqrt[3]{-frac{q}{2}-sqrt{frac{q^{2}}{4}+frac{p^{3}}{27}}} & left(-frac{1}{2}-ifrac{sqrt{3}}{2} right)sqrt[3]{-frac{q}{2}+sqrt{frac{q^{2}}{4}+frac{p^{3}}{27}}}+left(-frac{1}{2}+ifrac{sqrt{3}}{2} right)sqrt[3]{-frac{q}{2}-sqrt{frac{q^{2}}{4}+frac{p^{3}}{27}}} end{align} right.

and because x=t-frac{a}{3} the three possible values of x are

x=left{ begin{align} & sqrt[3]{-frac{q}{2}+sqrt{frac{q^{2}}{4}+frac{p^{3}}{27}}}+sqrt[3]{-frac{q}{2}-sqrt{frac{q^{2}}{4}+frac{p^{3}}{27}}}-frac{a}{3} & left(-frac{1}{2}+ifrac{sqrt{3}}{2} right)sqrt[3]{-frac{q}{2}+sqrt{frac{q^{2}}{4}+frac{p^{3}}{27}}}+left(-frac{1}{2}-ifrac{sqrt{3}}{2} right)sqrt[3]{-frac{q}{2}-sqrt{frac{q^{2}}{4}+frac{p^{3}}{27}}}-frac{a}{3} & left(-frac{1}{2}-ifrac{sqrt{3}}{2} right)sqrt[3]{-frac{q}{2}+sqrt{frac{q^{2}}{4}+frac{p^{3}}{27}}}+left(-frac{1}{2}+ifrac{sqrt{3}}{2} right)sqrt[3]{-frac{q}{2}-sqrt{frac{q^{2}}{4}+frac{p^{3}}{27}}}-frac{a}{3} end{align} right.

and the cubic is solved whether D=frac{q^{2}}{4}+frac{p^{3}}{27} is positive, negative or zero.

If D is positive then there is one real and two complex roots.

If D is negative then there are three real roots.

If D = 0 then there is one real root (a triple root) or two real roots (a single and a double root.)

Lagrange resolvents

The symmetric group S3 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+isqrt{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 si 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,bright)^{3}.
The roots of this quadratic equation are
frac92,ab-{a}^{3}- frac{27}{2},c pm frac{1}{2},sqrt{D},
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
left (x-rright )left (x^2+(a+r)x+b+ar+r^2 right ) = x^3+ax^2+bx+c.
Hence if we know one root we can find the other two by solving a quadratic equation, giving
frac12 left(-a-r pm sqrt{-3r^2-2ar+a^2-4b}right)
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

f(x) = ax^3 + bx^2 + cx + d = a(x - x_1)(x - x_2)(x - x_3),
with a,b,c,d in Bbb{R} and a neq 0, let
q = frac{3ac-b^2}{9a^2}

and

r = frac{9abc - 27a^2d - 2b^3}{54a^3},

we define the discriminant:

Delta = q^3+r^2.,

There are two distinct cases:

  • Delta geq 0,

in which case there is one real root and 2 imaginary. We define:
s = sqrt[3]{r + sqrt{q^3+r^2}}
and
t = sqrt[3]{r - sqrt{q^3+r^2}}.

  • Delta < 0,

in which case we have 3 real roots. We express the complex quantity r + isqrt{-Delta} in polar form:
r + isqrt{-Delta} = (rho, theta),
rho = sqrt{r^2-Delta},
theta = acos(frac{r}{rho}),
and we define:
s = (sqrt[3]{rho}, frac{theta}{3}),
and
t = (sqrt[3]{rho}, -frac{theta}{3}).,

In both cases, the solutions are

x_1 = s+t-frac{b}{3a},
x_2 = -frac{1}{2}(s+t)-frac{b}{3a}+frac{sqrt{3}}{2}(s-t)i,
x_3 = -frac{1}{2}(s+t)-frac{b}{3a}-frac{sqrt{3}}{2}(s-t)i.

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_{1over3} as
r_0 = sqrt{p},C_{1over3}(p^{-frac{3}{2}}q),,
r_1 = -sqrt{p},C_{1over3}(-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 = ta/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_{1over3}(t_{a;b;c})-{aover 3} ,,
r_1 = -sqrt{p},C_{1over3}(-t_{a;b;c})-{aover 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_{1over3}(t) is the sole real root, or t < −2 and -C_{1over3}(-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_{1over3}(-it)-iC_{1over3}(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_{1over3}(t) = iC_{1over3}(-it)-iC_{1over3}(it) = 2 operatorname{sinh}left(operatorname{arcsinh}left({tover2}right)/3right),,

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.

Derivative

The derivative f'(x)=3ax^2+2bx+c, will yield x=frac{-b pm sqrt {b^2-3ac }}{3a} when f'(x)=0,. Bearing its resemblance to the quadratic formula, this formula can be used to find the critical points of a cubic function. It turns out that, if b^2-3ac > 0, , then the cubic function will have two critical points — a local maximum and a local minimum; if b^2-3ac = 0, , then there is one critical point, and it will yield the inflection point; if b^2-3ac < 0,, then there are no critical points.

Bipartite cubics

The graph of

y^2 = x(x-a)(x-b),

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

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

f(x) = sqrt{x(x-a)(x-b)},

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

(0,a) cup (b,+infty).,

See also

Notes

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.

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