Quintic equation

In mathematics, a quintic equation is a polynomial equation in which the greatest exponent on the independent variable is five. For example:

Related Topics:
Mathematics - Polynomial - Equation

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:x^5-4x^4+2x^3-3x+7=0,

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Finding the zeroes of a polynomial — values of x which satisfy such an equation — given its coefficients was long a prominent mathematical problem. The linear and quadratic cases fell fairly quickly; after a while cubic and quartic

Related Topics:
Linear - Quadratic - Cubic - Quartic

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succumbed. But if there was some pattern to the formulae no one could see it, and the quintic was proving to be extremely stubborn.

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Eventually, Paolo Ruffini and Niels Henrik Abel were able to prove that there is no single finite expression of +, -, ×, ÷, and

Related Topics:
Paolo Ruffini - Niels Henrik Abel - Prove - Expression

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radicals that can produce them from the coefficients for all quintics. This is sometimes, mistakenly, taken to mean that there is no algebraic solution to the general quintic, which is false. Such a solution is given below. It should be noted however that numerical methods such as

Related Topics:
Radicals - Algebraic solution

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Newton's method

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give excellent results if all we require are numerical values for the roots, and that various transcendental functions such as the theta function or the Dedekind eta function can be used to give closed expressions.

Related Topics:
Theta function - Dedekind eta function

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The honour of proving the quartic formula to be the last of its kind, i.e. that there was no solution in radicals to the general sextic, septic, octic, formula, and so on, fell to Evariste Galois, who had an ingenious

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insight which reduced the issue to a question of group theory.

Related Topics:
Insight - Group theory

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