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| {{distinguish|Binomial distribution}}
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| {{Otheruses}}
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| {{refimprove|date=March 2011}}
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| In [[algebra]], a '''binomial''' is a [[polynomial]] with two terms<ref>{{Cite web
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| | last = Weisstein
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| | first = Eric
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| | authorlink = Eric Weisstein
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| | coauthors =
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| | title = Binomial
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| | work =
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| | publisher = Wolfram MathWorld
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| | date =
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| | url = http://mathworld.wolfram.com/Binomial.html
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| | format =
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| | doi =
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| | accessdate = 29 March 2011}}</ref> —the sum of two [[monomial]]s—often bound by parentheses or brackets when operated upon. It is the simplest kind of polynomial after the [[monomial]]s.
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| ==Operations on simple binomials==
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| *The binomial <math> a^2 - b^2 </math> can be factored as the product of two other binomials.
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| ::<math> a^2 - b^2 = (a + b)(a - b). </math>
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| :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>.
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| :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
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| *The product of a pair of linear binomials <math>(ax+b)</math> and <math>(cx+d)</math> is:
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| ::<math> (ax+b)(cx+d) = acx^2+adx+bcx+bd.</math>
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| *A binomial raised to the ''n<sup>th</sup>'' [[Exponentiation|power]], represented as
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| ::<math> (a + b)^n </math>
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| :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>. | |
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| *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>.
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| ==See also==
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| *[[Binomial theorem]]
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| *[[Completing the square]]
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| *[[Binomial distribution]]
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| *[[Binomial coefficient]]
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| *[[Binomial-QMF]] (Daubechies Wavelet Filters)
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| *The [[list of factorial and binomial topics]] contains a large number of related links.
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| *[[Binomial series]]
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| == Notes ==
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| {{reflist}}
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| ==References==
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| * L. Bostock, and S. Chandler (1978). Pure Mathematics 1. ISBN 0-85950-092-6. pp. 36
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| * {{springer|title=Binomial|id=p/b016400}}
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| [[Category:Algebra]]
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