# Binomial theorem

*13 January 2011*.

In elementary algebra, the **binomial theorem** or the binomial expansion is a mechanism by which expressions of the form (*x* + *y*)^{n} can be expanded. It is the identity that states that for any non-negative integer *n*,

where

is a binomial coefficient. Another useful way of stating it is the following:

## Contents |

### Pascal's triangle

An alternate way to find the binomial coefficients is by using Pascal's triange. The triangle is built from apex down, starting with the number one alone on a row. Each number is equal to the sum of the two numbers directly above it.

n=0 1 n=1 1 1 n=2 1 2 1 n=3 1 3 3 1 n=4 1 4 6 4 1 n=5 1 5 10 10 5 1

Thus, the binomial coefficients for the expression (*x* + *y*)^{4} are 1, 3, 6, 4, and 1.

## Proof

One way to prove this identity is by mathematical induction.

**Base case**: n = 0

**Induction case**: Now suppose that it is true for n : and prove it for n + 1.

and the proof is complete.

## Examples

These are the expansions from 0 to 6.

## Newton's binomial theorem

There is also **Newton's binomial theorem**, proved by Isaac Newton, that goes beyond elementary algebra into mathematical analysis, which expands the same sum (*x* + *y*)^{n} as an infinite series when *n* is not an integer or is not positive.

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