# Monomial basis

In mathematics the monomial basis of a polynomial ring is its basis (as vector space or free module over the field or ring of coefficients) that consists in the set of all monomials. In fact, a polynomial may be uniquely written as a linear combination of monomials.

Univariate polynomials expressed on the monomial basis can be evaluated efficiently using Horner's method.

## Definition

The monomial basis for the vector space ${\displaystyle \Pi _{n}}$ of polynomials with degree n is the polynomial sequence of monomials

${\displaystyle 1,x,x^{2},.\ldots ,x^{n}}$

The monomial form of a polynomial ${\displaystyle p\in \Pi _{n}}$ is a linear combination of monomials

${\displaystyle a_{0}1+a_{1}x+a_{2}x^{2}+\ldots +a_{n}x^{n}}$

alternatively the shorter sigma notation can be used

${\displaystyle p=\sum _{\nu =0}^{n}a_{\nu }x^{\nu }}$

## Notes

A polynomial can always be converted into monomial form by calculating its Taylor expansion around 0.

## Examples

${\displaystyle 1+x+3x^{4}}$