binomial theorem (redirected from binomial theorems)

binomial theorem

In algebra, a formula for expansion of the binomial (x + y) raised to any positive integer power. A simple case is the expansion of (x + y)², which is x² + 2xy + y². In general, the expression (x + y)ⁿ expands to the sum of (n + 1)terms in which the power of x decreases from n to 0 while the power of y increases from 0 to n in successive terms. The terms can be represented in factorial notation by the expression [n!/((n − r)!r!)]x^{n − r}y^r in which r takes on integer values from 0 to n.

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binomial theorem

a mathematical theorem that gives the expansion of any binomial raised to a positive integral power, n. It contains n + 1 terms: (x + a)ⁿ = xⁿ + nx^n--1 a + [n(n--1)/2] x^n--2a² +…+ (ⁿ_k) x^n--ka^k + … + aⁿ, where (ⁿ_k) = n!/(n--k)!k!, the number of combinations of k items selected from n - cf CED3 and CED4 for super/subscript chars

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binomial theorem [bī′nō·mē·əl ′thir·əm]

(mathematics)

The rule for expanding (x + y)ⁿ.