binomial theorem

Math.

the theorem giving the expansion of a binomial raised to any power.

[1865-70]

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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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▪ mathematics

      statement that, for any positive integer n, the nth power of the sum of two numbers a and b may be expressed as the sum of n + 1 terms of the form

      in the sequence of terms, the index r takes on the successive values 0, 1, 2, . . . , n. The coefficients, called the binomial coefficients, are defined by the formula

      in which n! (called n factorial) is the product of the first n natural numbers 1, 2, 3, . . . , n (and where 0! is defined as equal to 1). The coefficients may also be found in the array often called Pascal's triangle

      by finding the rth entry of the nth row (counting begins with a zero in both directions). Each entry in the interior of Pascal's triangle is the sum of the two entries above it.

      The theorem is useful in algebra as well as for determining permutations, combinations, and probabilities. For positive integer exponents, n, the theorem was known to Islamic and Chinese mathematicians of the late medieval period. Isaac Newton (Newton, Sir Isaac) stated in 1676, without proof, the general form of the theorem (for any real number n), and a proof by Jakob Bernoulli (Bernoulli, Jakob) was published in 1713, after Bernoulli's death. The theorem can be generalized to include complex exponents, n, and this was first proved by Niels Henrik Abel (Abel, Niels Henrik) in the early 19th century.

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