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binomial

binomial (bī"nō'mēul) [key], polynomial expression (see polynomial) containing two terms, for example, x+y. The binomial theorem, or binomial formula, gives the expansion of the nth power of a binomial (x+y) for n=1, 2, 3, … , as follows:

(x+y)n = xn + (n/1)xn-1y + n(n-1)/(2!) xn-2 y2 + n(n-1)(n-2)/(3!)) xn-3 y3 + ... + nxyn-1 + yn

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=x5+5x4y+10x3y2+10x2y3+5xy4+y5. 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:

1

1 1

1 2 1

1 3 3 1

1 3 6 4 1

1 5 10 10 5 1

The Columbia Electronic Encyclopedia, 6th ed. Copyright © 2007, Columbia University Press. All rights reserved.

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