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 $\Pi _{n}$ 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 $\Pi _{4}$

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

See also

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