Editing Quartic function (section)

{{short description|Polynomial function of degree 4}}
{{Distinguish|Quantic (disambiguation){{!}}Quantic}}
{{about|the univariate case|the bivariate case|Quartic plane curve}}
{{redirect|Biquadratic function|the use in computer science|Biquadratic rational function}}
{{Use dmy dates|date=December 2017}}
[[File:Polynomialdeg4.svg|thumb|right|233px|Graph of a polynomial of degree 4, with 3 [[critical point (mathematics)|critical points]] and four [[real number|real]] [[root of a polynomial|roots]] (crossings of the ''x'' axis) (and thus no [[complex number|complex]] roots). If one or the other of the local [[minimum|minima]] were above the ''x'' axis, or if the local [[maximum]] were below it, or if there were no local maximum and one minimum below the ''x'' axis, there would only be two real roots (and two complex roots). If all three local extrema were above the ''x'' axis, or if there were no local maximum and one minimum above the ''x'' axis, there would be no real root (and four complex roots).  The same reasoning applies in reverse to polynomial with a negative quartic coefficient.]]

In [[algebra]], a '''quartic function'''  is a [[function (mathematics)|function]] of the form{{ref|Alpha|α}}
:<math>f(x)=ax^4+bx^3+cx^2+dx+e,</math>
where ''a'' is nonzero,
which is defined by a [[polynomial]] of [[Degree of a polynomial|degree]] four, called a '''quartic polynomial'''.

A ''[[quartic equation]]'', or equation of the fourth degree, is an equation that equates a quartic polynomial to zero, of the form
:<math>ax^4+bx^3+cx^2+dx+e=0 ,</math>
where {{nowrap|''a'' ≠ 0}}.<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Quartic Equation|url=https://mathworld.wolfram.com/QuarticEquation.html|access-date=2020-07-27|website=mathworld.wolfram.com|language=en}}</ref>
The [[derivative]] of a quartic function is a [[cubic function]].

Sometimes the term '''biquadratic''' is used instead of ''quartic'', but, usually, '''biquadratic function''' refers to a [[quadratic function]] of a square (or, equivalently, to the function defined by a quartic polynomial without terms of odd degree), having the form
:<math>f(x)=ax^4+cx^2+e.</math>

Since a quartic function is defined by a polynomial of even degree, it has the same infinite limit when the argument goes to positive or negative [[infinity]]. If ''a'' is positive, then the function increases to positive infinity at both ends; and thus the function has a [[Maxima and minima|global minimum]]. Likewise, if ''a'' is negative, it decreases to negative infinity and has a global maximum. In both cases it may or may not have another local maximum and another local minimum.

The degree four (''quartic'' case) is the highest degree such that every polynomial equation can be solved by [[Nth root|radicals]], according to the [[Abel–Ruffini theorem]].