binomial theorem

The binomial theorem is a method of expanding the binomial expression (x + y)n into a finite or infinite series of powers of x and y, where n is a number either integral or fractional, positive or negative, rational or irrational; thus


(x + y)n = xn + an-1xn-1y + an-2xn-2y2 + ... + yn


The coefficients ai are called binomial coefficients.


The binomial theorem was discovered by Isaac 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.