Polynomial

A polynomial in one variable Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} is a function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} of the form:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} that appears in the polynomial is called the degree of the polynomial.

• Example: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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]}