Quintic function

Polynomial of degree 5: f(x) = (x+4)(x+2)(x+1)(x-1)(x-3)/20+2

In mathematics, a quintic equation is a polynomial equation in which the greatest exponent on the independent variable is five. It is of the form:

$ax^{5}+bx^{4}+cx^{3}+dx^{2}+ex+f=0,$

where a, b, c, d, e, and f are members of a field, (typically the rational numbers, the real numbers or the complex numbers), and $a\neq 0$.

Because they have an odd degree, normal quintic functions appear similar to normal cubic functions when graphed, except they may possess an additional local maximum and minimum each. The derivative of a quintic function is a quartic function.

Finding roots of a quintic equation

Finding the roots of a polynomial — values of x that satisfy such an equation — in the rational case given its coefficients has been a prominent mathematical problem.

Solving linear, quadratic, cubic and quartic equations by factorization into radicals is fairly straightforward when the roots are rational and real; there are also formulae that yield the required solutions. However, there is no formula for general quintic equations over the rationals in terms of radicals; this was first proved by the Abel-Ruffini theorem, first published in 1824, which was one of the first applications of group theory in algebra. This result also holds for equations of higher degrees.

Some fifth degree equations can be solved by factorizing into radicals, for example x⁵ − x⁴ − x + 1 = 0, which can be written as (x² + 1)(x + 1)(x − 1)² = 0. Other quintics like x⁵ − x + 1 = 0 cannot be easily factorized and solved in this manner. Évariste Galois developed techniques for determining whether a given equation could be solved by radicals which gave rise to the field of Galois theory. Using Galois theory, John Stuart Glashan, George Paxton Young, and Carl Runge showed in 1885 that any irreducible solvable quintic in Bring-Jerrard form,

$x^{5}+ax+b=0$

must have the following form:

$x^{5}+{\frac {5\mu ^{4}(4\nu +3)}{\nu ^{2}+1}}x+{\frac {4\mu ^{5}(2\nu +1)(4\nu +3)}{\nu ^{2}+1}}=0$

where $\mu$ and $\nu$ are rational. In 1994, Spearman and Williams gave an alternative,

$x^{5}+{\frac {5e^{4}(3-4c\epsilon )}{c^{2}+1}}x+{\frac {-4e^{5}(11\epsilon +2c)}{c^{2}+1}}=0$

for $\epsilon =\pm 1$. Since by judicious use of Tschirnhaus transformations it is possible to transform any quintic into Bring-Jerrard form, this gives a necessary and sufficient condition for a quintic to be solvable in radicals. The relationship between the 1885 and 1994 parametrizations can be seen by defining the expression

$b=(4/5)(a+20+2{\sqrt {(20-a)(5+a)}})$

where

$a={\frac {5(4v+3)}{v^{2}+1}}$

and using the negative case of the square root yields, after scaling variables, the first parametrization while the positive case gives the second with $\epsilon =-1$. It is then a necessary (but not sufficient condition) that the irreducible solvable quintic

$z^{5}+a\mu ^{4}z+b\mu ^{5}=0$

with rational coefficients must satisfy the simple quadratic curve

$y^{2}=(20-a)(5+a)$

for some rational a,y.

There also exist other methods of solving quintics. About 1835, Jerrard showed that quintics can be solved by using ultraradicals (also known as Bring radicals), the real roots of t⁵ + t − a for real numbers a. In 1858 Charles Hermite showed that the Bring radical could be characterized in terms of the Jacobi theta functions and their associated elliptic modular functions, using an approach similar to the more familiar approach of solving cubic equations by means of trigonometric functions. At around the same time, Leopold Kronecker, using group theory developed a simpler way of deriving Hermite's result, as had Francesco Brioschi. Later, Felix Klein came up with a particularly elegant method that relates the symmetries of the icosahedron, Galois theory, and the elliptic modular functions that feature in Hermite's solution, giving an explanation for why they should appear at all, and develops his own solution in terms of generalized hypergeometric functions.

Numerical methods such as the Newton-Raphson method or trial and error give results very quickly if only approximate numerical values for the roots are required, or if it is known that the solutions comprise only simple expressions (such as in exams). Other methods such as Laguerre's method or the Jenkins-Traub method may also be used to more reliably find the roots of quintic equations numerically.

References

• Charles Hermite, "Sur la Résolution de L'Equation Du Cinquème Degré" (1858) in Œuvres de Charles Hermite, t.2, pp. 5-21, Gauthier-Villars, 1908.
• Felix Klein, Lectures on the Icosahedron and the Solution of Equations of the Fifth Degree, trans. George Gavin Morrice, Trübner & Co., 1888. ISBN 0-486-49528-0.
• Ian Stewart, Galois Theory 2nd Edition, Chapman and Hall, 1989. ISBN 0-412-34550-1. Discusses Galois Theory in general including a proof of insolvability of the general quintic.

See also

• Solvable group
• Theory of equations

External links

• Mathworld - Quintic Equation - more details on methods for solving Quintics.
• Solving the Quintic with Mathematica - poster on Quintic solutions
• Lectures on the Icosahedron - Klein's book is available online