Binomial — Chambers's Encyclopaedia

Binomial, in Algebra, is an expression which consists of two terms, separated by + or -, such as $x+y$ , $ab-cd$ . Monomial, trinomial, polynomial are the names given to expressions consisting of one, three, more than three terms. The Binomial Theorem is a method of expanding $(x+y)^n$ into a finite or an infinite series of powers of $x$ and $y$ , when $n$ is a number either integral or fractional, positive or negative, rational or irrational; thus $(x+y)^n=$

x^n + nx^{n-1}y + \frac{n(n-1)}{1.2}x^{n-2}y^2 + \frac{n(n-1)(n-2)}{1.2.3}x^{n-3}y^3 + \dots

It was discovered by Newton about 1666, and was first published in 1704 in the second appendix to Newton's Optics. That particular case of the theorem when $n$ is a positive integer was known to mathematicians prior to Newton (e.g. Briggs and Pascal), and Newton himself gave no demonstration of the truth of his theorem. Many demonstrations have been given by subsequent mathematicians, but the validity of most of them has been called in question.

For more information regarding the Binomial Theorem, its applications and its developments, see Chrystal's Text-Book of Algebra.