In algebra, a binomial is a polynomial that is the sum of two terms, each of which is a monomial. It is the simplest kind of a sparse polynomial after the monomials.

Definition

A binomial is a polynomial which is the sum of two monomials. A binomial in a single indeterminate (also known as a univariate binomial) can be written in the form where a and b are numbers, and m and n are distinct non-negative integers and x is a symbol which is called an indeterminate or, for historical reasons, a variable. In the context of Laurent polynomials, a Laurent binomial, often simply called a binomial, is similarly defined, but the exponents m and n may be negative. More generally, a binomial may be written as:

Examples

Operations on simple binomials

x2 − y2 , the difference of two squares, can be factored as the product of two other binomials: (ax + b) and (cx + d ) is a trinomial: n th power, represented as (x + y)n can be expanded by means of the binomial theorem or, equivalently, using Pascal's triangle. For example, the square (x + y)2 of the binomial (x + y) is equal to the sum of the squares of the two terms and twice the product of the terms, that is: n th power uses the numbers n rows down from the top of the triangle. (m, n) -formula" for generating Pythagorean triples: m < n , let , , and a2 + b2 = c2 .