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{{distinguish|Binomial distribution}}
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{{Otheruses}}
{{refimprove|date=March 2011}}
 
In [[algebra]], a '''binomial''' is a [[polynomial]] with two terms<ref>{{Cite web
  | last = Weisstein
  | first = Eric
  | authorlink = Eric Weisstein
  | coauthors =
  | title = Binomial
  | work =
  | publisher = Wolfram MathWorld
  | date =
  | url = http://mathworld.wolfram.com/Binomial.html
  | format =
  | doi =
  | accessdate = 29 March 2011}}</ref> &mdash;the sum of two [[monomial]]s&mdash;often bound by parentheses or brackets when operated upon. It is the simplest kind of polynomial after the [[monomial]]s.
 
==Operations on simple binomials==
*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>.
:This can also be extended to <math> a^2 + b^2 = a^2 - (ib)^2 = (a - ib)(a + ib) </math> when working over the complex numbers
 
*The product of a pair of linear binomials <math>(ax+b)</math> and <math>(cx+d)</math> is:
::<math> (ax+b)(cx+d) = acx^2+adx+bcx+bd.</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]]. Taking a simple example, the [[perfect square]] binomial <math>(p+q)^2</math> 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 <math>p^2+2pq+q^2</math>.
 
*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</math>, <math>b=2mn</math>, <math>c=n^2+m^2</math>, then <math>a^2+b^2=c^2</math>.
 
==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]]
 
== Notes ==
{{reflist}}
 
==References==
* L. Bostock, and S. Chandler (1978). Pure Mathematics 1. ISBN 0-85950-092-6. pp. 36
* {{springer|title=Binomial|id=p/b016400}}
 
[[Category:Algebra]]

Revision as of 16:36, 4 February 2014

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