Binomial

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.

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 $a^2+b^2=c^2$.

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.

External links

* [http://www.allmathwords.org/binomial.html Binomial] All Math Words Encyclopedia for grades 7-10.