Monomial basis

In mathematics a monomial basis is a way to uniquely describe 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 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_{ u=0}^n a_{ u}x^ u

=Notes=

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

=Examples=

A polynomial in Pi_4

:1+x+3x^4

=See also=

*Polynomial sequence *Newton polynomial *Lagrange polynomial *Bernstein form *Chebyshev form