The binomial theorem is an important formula about the expansion of powers of sums. Its simplest version reads
is any non-negative integer and the numbers
are the binomial coefficients
. This formula, and the triangular arrangement
of the binomial coefficients, are often attributed to Blaise Pascal
who described them in the 17th century
. It was however known long before to Chinese mathematicians.
The cases n=2, n=3 and n=4 are the ones most commonly used:
- (x + y)2 = x2 + 2xy + y2
- (x + y)3 = x3 + 3x2y + 3xy2 + y3
- (x + y)4 = x4 + 4x3y + 6x2y2 + 4xy3 + y4
Formula (1) is valid for all real
, and more generally for any elements x
of a ring
as long as xy
Isaac Newton generalized the formula to other exponents by considering an infinite series:
can be any complex number
(in particular r
can be any real number, not necessarily positive and not necessarily an integer), and the coefficients are given by
(which in case k
= 0 is a product of no numbers at all
and therefore equal to 1, and in case k
= 1 is equal to r
, as the additional factors (r
- 1), etc., do not appear in that case).
The sum in (2) converges and the equality is true whenever the real or complex numbers x and y are "close together" in the sense that the absolute value |x/y| is less than one.
The geometric series is a special case of (2) where we choose y = 1 and r = -1.
Formula (2) is also valid for elements x and y of a Banach algebra as long as xy = yx, y is invertible and ||x/y|| < 1.
The binomial theorem can be stated by saying that the polynomial sequence
is of binomial type