# binomial

**binomial**bī˝nō´mēəl [key], polynomial expression (see polynomial ) containing two terms, for example,

*x*+

*y.*The binomial theorem, or binomial formula, gives the expansion of the

*n*th power of a binomial (

*x*+

*y*) for

*n*=1, 2, 3, … , as follows: where the ellipsis (…) indicates a continuation of terms following the same pattern. For example, using the formula and reducing fractions, one obtains (

*x*+

*y*)

^{5}=

*x*

^{5}+5

*x*

^{4}

*y*+10

*x*

^{3}

*y*

^{2}+10

*x*

^{2}

*y*

^{3}+5

*x*

*y*

^{4}+

*y*

^{5}. The coefficients 1,

*n, n*(

*n*−1)/1·2, etc., of

*x*and

*y*may also be found from an array known as Pascal's triangle (for Blaise Pascal ), formed by adding adjacent numbers to find the number below them as follows:

*The Columbia Electronic Encyclopedia,* 6th ed. Copyright © 2012, Columbia University Press. All rights reserved.

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