Quintic equation

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

Algebraic solution of the general quintic

We now may express the roots of any polynomial

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:x^5 + px +q,

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in terms of the Bring radical as

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:left(-rac{p}{4} ight)^rac{1}{4}operatorname{BR}left(rac{(-5/p)^rac{5}{4} q}{4} ight)

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and its four conjugates. We have a reduction to the Bring-Jerrard form in terms of solvable polynomial equations, and we used transformations involving polynomial expressions in the roots only up to the fourth degree, which means inverting the transformation may be done by finding the roots of a polynomial solvable in radicals. This procedure produces extraneous solutions, but when we have found the correct ones by numerical means we can also write down the roots of the quintic in terms of square roots, cube roots, and the Bring radical, which is therefore an algebraic solution in terms of algebraic functions of a single variable — an algebraic solution of the general quintic.

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