We know that

For higher powers, the expansion gets very tedious by hand! Fortunately, the Binomial Theorem gives us the expansion for any positive integer power of (x+y):

Binomial Theorem

For any positive integer n,

Proof by Induction

Combinatorial Proof

Connection to Pascal's Triangle

Example

By the Binomial Theorem,

Extensions of the Binomial Theorem

A useful special case of the Binomial Theorem is

This formula can be extended to all real powers a:

Notice that the formula now gives an infinite series. (When a = n is a positive integer, all but the first (n+1) terms are 0 since after this n-n ( = 0) appears in each numerator.)

This expansion is very useful for approximating (1+x)^a for |x| << 1:

But for |x| << 1, higher powers of x get small very quickly, so (1+x)^a can be approximated to any accuracy we need by truncating the series after a finite number of terms.

Example

For |x| << 1,

This type of reasoning is useful in investigating what happens when a physical system is perturbed slightly, introducing a new very small term x.

Binomial Theorem

For any positive integer n,