Binomial

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For other uses, see Binomial (disambiguation).

In elementary algebra, a binomial is a polynomial with two terms: the sum of two monomials. It is the simplest kind of polynomial except for a monomial.

The binomial $a 2 - b 2$ can be factored as the product of two other binomials:

a 2 - b 2 = (a + b)(a - b).

(This is a special case of the more general formula $a^{n+1} - b^{n+1} = (a - b)\sum_{k=0}^{n} a^{k}\,b^{n-k}$ .)

The product of a pair of linear binomials a x + b and c x + d is:

(a x + b)(c x + d) = a c x 2 + (a d + b c) x + b d .

A binomial raised to the n^th power, represented as

(a + b) n

can be expanded by means of the binomial theorem or, equivalently, using Pascal's triangle.

[edit] Example

A simple but interesting application of the cited binomial formula is the "(m,n)-formula" for generating Pythagorean triples: for m < n, let $a=n^2-m^2,\ b=2mn,\ c=n^2+m^2$ , then $a 2 + b 2 = c 2$ .