binomial

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.

Operations on simple binomials

• The binomial a² − b² can be factored as the product of two other binomials:

a² − b² = (a + b)(a − b).

This is a special case of the more general formula: .

• The product of a pair of linear binomials (ax + b) and (cx + d) is:

(ax + b)(cx + d) = acx² + axd + bcx + bd.

• 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 term, adding twice the product of the first and second terms and finally adding the square of the second term, to give p² + 2pq + q².

• A simple but interesting application of the cited binomial formula is the "(m,n)-formula" for generating Pythagorean triples: for m < n, let a = n² − m², b = 2mn, c = n² + m², then a² + b² = c².

See also

• Completing the square
• Binomial distribution
• Binomial coefficient
• Binomial-QMF (Daubechies Wavelet Filters)
• The list of factorial and binomial topics contains a large number of related links.
• Binomial series