Quartic function

In algebra, a fourth degree polynomial is a polynomial of the form

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

with not equal to zero. A quartic function is the mapping corresponding to this polynomial . $f\colon \R\to \R$ A biquadratic function is a quartic function with b=0 and d=0 .

A quartic equation or fourth degree equation is an equation of the form

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

with $a\not =0$ . Accordingly, one also speaks of biquadratic equations.

Properties of quartic functions

In the following, let $f\colon \R\to \R$ a quartic function defined $a\not =0$ by $f(x)=ax^{4}+bx^{3}+cx^{2}+dx+e,$ with

Behaviour at infinity

As with all integer functions of even degree, the following applies

$\lim \limits _{{x\to +\infty }}f(x)=+\infty$ , $\lim \limits _{{x\to -\infty }}f(x)=+\infty$ ,

if the leading coefficient is positive, and

$\lim \limits _{{x\to +\infty }}f(x)=-\infty$ , $\lim \limits _{{x\to -\infty }}f(x)=-\infty$ ,

if is negative.

Nulls

A fourth degree polynomial has at most four zeros, but can also have no real zeros. If zeros are counted according to their multiplicity, it has exactly four complex zeros. If all zeros are real, the discriminant is nonnegative. The converse does not hold, the polynomial $x^{4}+4$ has positive discriminant but no real zeros.

For the (complex) zeros there is a solution formula, see quartic equation. The numerical finding of real zeros is possible, for example, with Newton's method.

Local extremes

As a polynomial function, differentiable arbitrarily many times; for its 1st derivative , we get the cubic function

$f'(x)=4ax^{3}+3bx^{2}+2cx+d$ .

If its discriminant is positive, then has exactly three local extrema, namely for a>0 one local maximum and two local minima or for a<0 two local maxima and one local minimum.

Turning Points

A quartic function has at most two inflection points $(x_{W};f(x_{W}))$ . The inflection points $x_{W}$ are the zeros of the 2nd derivative $f''(x)=12ax^{2}+6bx+2c$ .

Fourth degree polynomials

Let be any ring. Fourth degree polynomials over are expressions of the form

$ax^{4}+bx^{3}+cx^{2}+dx+e\in R[x]$

with $a,b,c,d,e\in R$ and $a\not =0$ . Formally, these are elements of the polynomial ring of degree 4, they define mappings from to . For $R=\mathbb {R}$ are quartic functions in the above sense.

If is an algebraically closed body, every fourth degree polynomial decays as a product of four linear factors.

More generally, quartic polynomials in variables are expressions of the form

$\sum _{i,j,k,l=1}^{n}a_{i,j,k,l}x_{i}x_{j}x_{k}x_{l}+\sum _{i,j,k=1}^{n}b_{i,j,k}x_{i}x_{j}x_{k}+\sum _{i,j=1}^{n}c_{i,j}x_{i}x_{j}+\sum _{i=1}^{n}d_{i}x_{i}+e\in R[x_{1},\ldots ,x_{n}]$ ,

Where not all $a_{i,j,k,l}$ are said to be zero. These polynomials define mappings from $R^{n}$ to . Their sets of zeros in $R^{n}$ are called quartic curves for n=2 and quartic surfaces for . n=3