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.
This is a
special case of the more general formula:
When working over the
complex numbers, this can also be extended to:
The product of a pair of linear binomials (ax + b) and (cx + d ) is a
trinomial:
A binomial raised to the nthpower, 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:
The numbers (1, 2, 1) appearing as multipliers for the terms in this expansion are the
binomial coefficients two rows down from the top of Pascal's triangle. The expansion of the nth power uses the numbers n rows down from the top of the triangle.
An application of the above formula for the square of a binomial is the "(m, n)-formula" for generating
Pythagorean triples:
For m < n, let a = n2 − m2, b = 2mn, and c = n2 + m2; then a2 + b2 = c2.
Binomials that are sums or differences of
cubes can be factored into smaller-
degree polynomials as follows: