Binomial
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2008-07-11T18:44:34Z
128.235.72.163
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In [[elementary algebra]], a '''binomial''' is a [[polynomial]] with two terms: the sum of two [[monomial]]s. It is the simplest kind of polynomial except for a monomial.
The binomial <math> a^2 - b^2 </math> can be factored as the product of two other binomials:
:<math> a^2 - b^2 = (a + b)(a - b). </math>
(This is a special case of the more general formula <math> a^{n+1} - b^{n+1} = (a - b)\sum_{k=0}^{n} a^{k}\,b^{n-k}</math>.)
The product of a pair of linear binomials ''a x + b'' and ''c x + d'' is:
:<math> (a x + b)(c x + d) = a c x^2 + (a d + b c) x + b d.</math>
A binomial raised to the ''n<sup>th</sup>'' [[Exponentiation|power]], represented as
:<math> (a + b)^n </math>
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 triple]]s: for ''m < n'', let <math>a=n^2-m^2,\ b=2mn,\ c=n^2+m^2</math>, then <math>a^2+b^2=c^2</math>.
== 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.
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