Binomial

From Wikipedia, the free encyclopedia

For other uses, see Binomial (disambiguation).

In elementary algebra, a binomial is a polynomial with two terms—the sum of two monomials—often bound by parenthesis or brackets when operated upon. It is the simplest kind of polynomial other than monomials.

[edit] Operations on simple binomials

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 x 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. Taking a simple example, the perfect square binomial

(p + q) 2

can be found by squaring the :first digit, adding twice the product of the first and second digit and finally adding the square of the second digit, to give

p 2 + 2 p q + q 2

.

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