Monomial basis

In mathematics a monomial basis is a way to describe uniquely a polynomial using a linear combination of monomials. This description, the monomial form of a polynomial, is often used because of the simple structure of the monomial basis.

Polynomials in monomial form can be evaluated efficiently using the Horner algorithm.

Definition

The monomial basis for the vector space Π_n of polynomials with degree n is the polynomial sequence of monomials

$1,x,x^2,.\ldots,x^n$

The monomial form of a polynomial $p \in \Pi_n$ is a linear combination of monomials

$a_0 1 + a_1 x + a_2 x^2 + \ldots + a_n x^n$

alternatively the shorter sigma notation can be used

$p=\sum_{\nu=0}^n a_{\nu}x^\nu$

Notes

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

Examples

A polynomial in Π₄

1 + x + 3x⁴

See also

• Polynomial sequence
• Newton polynomial
• Lagrange polynomial
• Legendre polynomial
• Bernstein form
• Chebyshev form