In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

 a monomial basis is a way to describe uniquely a polynomial

Polynomial

In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...

 using a linear combination

Linear combination

In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results...

 of monomial

Monomial

In mathematics, in the context of polynomials, the word monomial can have one of two different meanings:*The first is a product of powers of variables, or formally any value obtained by finitely many multiplications of a variable. If only a single variable x is considered, this means that any...

s. 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 of polynomials with degree n is the polynomial sequence

Polynomial sequence

In mathematics, a polynomial sequence is a sequence of polynomials indexed by the nonnegative integers 0, 1, 2, 3, ..., in which each index is equal to the degree of the corresponding polynomial...

 of monomials


The monomial form of a polynomial is a linear combination of monomials


alternatively the shorter sigma notation can be used

See also

• Polynomial sequence
  Polynomial sequence
  In mathematics, a polynomial sequence is a sequence of polynomials indexed by the nonnegative integers 0, 1, 2, 3, ..., in which each index is equal to the degree of the corresponding polynomial...
  
  
• Newton polynomial
  Newton polynomial
  In the mathematical field of numerical analysis, a Newton polynomial, named after its inventor Isaac Newton, is the interpolation polynomial for a given set of data points in the Newton form...
  
  
• Lagrange polynomial
  Lagrange polynomial
  In numerical analysis, Lagrange polynomials are used for polynomial interpolation. For a given set of distinct points x_j and numbers y_j, the Lagrange polynomial is the polynomial of the least degree that at each point x_j assumes the corresponding value y_j...
  
  
• Legendre polynomial
• Bernstein form
• Chebyshev form