Polynomial

A polynomial in one variable $x$ is a function $f(x)$ of the form:

$f(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+...+a_{2}x^{2}+a_{1}x+a_{0}$

In elementary mathematics, the coefficients $a_{i}$ are typically chosen to be real or complex numbers. However, it makes sense to define a polynomial with coefficients in any ring.

The largest power of $x$ that appears in the polynomial is called the degree of the polynomial.

• Example: $f(x)=4x^{3}-3x+1$ is a degree 3 polynomial with integer coefficients.

A polynomial in two variables $x,y$ is, similarly, a finite sum

$\sum a_{ij}x^{i}y^{j}$, where the coefficients $a_{ij}$ are elements of some ring. Polynomials in 3 or more variables are defined similarly.

Some algorithms are said to perform in polynomial time. There are algorithms which can factor polynomials in polynomial time.^{[Citation Needed]}