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.

Contents

Definition

Notes

Examples

See also

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 be converted into monomial form by calculating its 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