What is "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.

   Not theories, but theorems (?), the intelligible products of contemplation, intellectual objects in the mind, and of and for the mind exclusively.
   --Coleridge.

   By the theorems, Which your polite and terser gallants practice, I re-refine the court, and civilize Their barbarous natures.
   --Massinger.

2. (Math.) A statement of a principle to be demonstrated.

   Note: A theorem is something to be proved, and is thus distinguished from a problem, which is something to be solved. In analysis, the term is sometimes applied to a rule, especially a rule or statement of relations expressed in a formula or by symbols; as, the binomial theorem; Taylor's theorem. See the Note under Proposition, n., 5.

   Binomial theorem. (Math.) See under Binomial.

   Negative theorem, a theorem which expresses the impossibility of any assertion.

   Particular theorem (Math.), a theorem which extends only to a particular quantity.

   Theorem of Pappus. (Math.) See Centrobaric method, under Centrobaric.

   Universal theorem (Math.), a theorem which extends to any quantity without restriction.

Binomial \Bi*no"mi*al\, a.

1. Consisting of two terms; pertaining to binomials; as, a binomial root.

2. (Nat. Hist.) Having two names; -- used of the system by which every animal and plant receives two names, the one indicating the genus, the other the species, to which it belongs.

   Binomial theorem (Alg.), the theorem which expresses the law of formation of any power of a binomial.

n. (context mathematics English) A formula giving the expansion of a binomial such as $(\; a\; +\; b\; )$ raised to any positive integer power, i.e. $(\; a\; +\; b\; )^\{n\}$. It's possible to expand the power into a sum of terms of the form $ax^\{b\}y^\{c\}$ where the coefficient of each term is a positive integer. For example:

n. a theorem giving the expansion of a binomial raised to a given power

In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the power into a sum involving terms of the form , where the exponents and are nonnegative integers with , and the coefficient of each term is a specific positive integer depending on and . For example,

(x + y) = x + 4xy + 6xy + 4xy + y.

The coefficient in the term of is known as the binomial coefficient $\tbinom nb$ or $\tbinom nc$ (the two have the same value). These coefficients for varying and can be arranged to form Pascal's triangle. These numbers also arise in combinatorics, where $\tbinom nb$ gives the number of different combinations of elements that can be chosen from an -element set.