Quintic equation

In algebra, a quintic function is a function 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, and $a$ is nonzero. In other words, a quintic function is defined by a polynomial of degree five.

If $a$ is zero but one of the coefficients $b$ , $c$ , $d$ , or $e$ is non-zero, the function is classified as either a quartic function, cubic function, quadratic function or linear function.

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 local minimum each. The derivative of a quintic function is a quartic function.

Setting $g (x) = 0$ and assuming $a \neq 0$ produces a quintic equation of the form:

Solving quintic equations in terms of radicals was a major problem in algebra, from the 16th century, when cubic and quartic equations were solved, until the first half of the 19th century, when the impossibility of such a general solution was proved, with the Abel–Ruffini theorem.

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