Binomial

Not to be confused with Binomial distribution.

For other uses, see Binomial (disambiguation).

In 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 after the monomials.

Operations on simple binomials

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

a² − b² = (a + b)(a − b).

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

This can also be extended to a² + b² = a² − (ib)² = (a − ib)(a + ib) when working over the complex numbers

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

(ax + b)(cx + d) = acx² + adx + 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

• Binomial theorem
• 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

References

• L. Bostock, and S. Chandler (1978). Pure Mathematics 1. ISBN 0 85950 0926. pp. 36.