Quintic equation

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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:

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

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.

1 Finding roots of a quintic equation
2 References
3 See also
4 External links

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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)²(x − 1) = 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.

There also exist other methods of solving quintics. Charles Hermite discovered that quintics can be solved by elliptic functions. 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. Such numbers can be expressed as infinite sequences, and are not all radical expressions. Various other transcendental functions such as the theta function or the Dedekind eta function can also be used to give closed expressions.

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).

In 1850 Kronecker's paper, On the Solution of the General Equation of the Fifth Degree, he solves the quintic equation by applying group theory.

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References

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.

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