Editing Quintic function (section)

{{short description|Polynomial function of degree 5}}
[[File:Quintic polynomial.svg|thumb|right|233px|Graph of a polynomial of degree 5, with 3 real zeros (roots) and 4 [[critical point (mathematics)|critical points]] ]]

In [[mathematics]], a '''quintic function''' is a [[function (mathematics)|function]] of the form

:<math>g(x)=ax^5+bx^4+cx^3+dx^2+ex+f,\,</math>

where {{mvar|a}}, {{mvar|b}}, {{mvar|c}}, {{mvar|d}}, {{mvar|e}} and {{mvar|f}} are members of a [[field (mathematics)|field]], typically the [[rational number]]s, the [[real number]]s or the [[complex number]]s, and {{mvar|a}} is nonzero. In other words, a quintic function is defined by a [[polynomial]] of [[Degree of a polynomial|degree]] five.

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

Setting {{math|''g''(''x'') {{=}} 0}} and assuming {{math|''a'' ≠ 0}} produces a '''quintic equation''' of the form:
:<math>ax^5+bx^4+cx^3+dx^2+ex+f=0.\,</math> 
Solving quintic equations in terms of [[Nth_root|radicals]] (''n''th roots) was a major problem in algebra from the 16th century, when [[cubic equation|cubic]] and [[quartic equation]]s 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]].