Quartic function

In algebra, a quartic function is a function of the form

${\displaystyle f(x)=ax^{4}+bx^{3}+cx^{2}+dx+e,}$^α

where a is nonzero, which is defined by a polynomial of 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

${\displaystyle ax^{4}+bx^{3}+cx^{2}+dx+e=0,}$

where a ≠ 0.^[1] 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

${\displaystyle f(x)=ax^{4}+cx^{2}+e.}$

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 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 radicals, according to the Abel–Ruffini theorem.