﻿

# binomial theorem

the theorem giving the expansion of a binomial raised to any power.
[1865-70]

* * *

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!)]xn -ryr in which r takes on integer values from 0 to n.

* * *

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.

* * *

Universalium. 2010.

### Look at other dictionaries:

• Binomial theorem — Theorem The o*rem, n. [L. theorema, Gr. ? a sight, speculation, theory, theorem, fr. ? to look at, ? a spectator: cf. F. th[ e]or[ e]me. See {Theory}.] 1. That which is considered and established as a principle; hence, sometimes, a rule. [1913… …   The Collaborative International Dictionary of English

• Binomial theorem — Binomial Bi*no mi*al, a. 1. Consisting of two terms; pertaining to binomials; as, a binomial root. [1913 Webster] 2. (Nat. Hist.) Having two names; used of the system by which every animal and plant receives two names, the one indicating the… …   The Collaborative International Dictionary of English

• binomial theorem — n. the general formula for the expansion of any binomial when raised to a power that is a positive whole number; the expansion of (a + b) n: discovered by Omar Khayyám and generalized by Sir Isaac Newton (Ex.: (a + b) 2 = a2 + 2ab + b2) …   English World dictionary

• Binomial theorem — The binomial coefficients appear as the entries of Pascal s triangle. In elementary algebra, the binomial theorem describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the power… …   Wikipedia

• binomial theorem — noun a theorem giving the expansion of a binomial raised to a given power • Topics: ↑statistics • Hypernyms: ↑theorem • Part Holonyms: ↑probability theory, ↑theory of probability …   Useful english dictionary

• binomial theorem — noun Date: 1753 a theorem that specifies the expansion of a binomial of the form (x + y)n as the sum of n + 1 terms of which the general term is of the form n!/((n k)! k!) x(n k) yk where k takes on values from 0 to n …   New Collegiate Dictionary

• binomial theorem — bino′mial the′orem n. math. the theorem giving the expansion of a binomial raised to any power • Etymology: 1865–70 …   From formal English to slang

• binomial theorem — noun A formula giving the expansion of a binomial such as raised to any positive integer power, i.e. . Its possible to expand the power into a sum of terms of the form where the coefficient of each term is a positive integer. For example …   Wiktionary

• binomial theorem — n. mathematical formula that provides the expansion of a binomial raised to any power …   English contemporary dictionary

• binomial theorem — noun a formula for finding any power of a binomial without multiplying at length …   English new terms dictionary