Divided differences

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 Horner's method.

Definition

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

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

The monomial form of a polynomial $p\in {\mathrm{\Pi }}_n$ is a linear combination of monomials

$a_{0}1+a_{1}x+a_2{x}^2+\dots +a_n{x}^n$

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 ${\mathrm{\Pi }}_4$

$1+x+3x^4$

See also

• Horner's method
• Polynomial sequence
• Newton polynomial
• Lagrange polynomial
• Legendre polynomial
• Bernstein form
• Chebyshev form