# Binomial theorem

The **binomial theorem**, also known as the binomial expansion, describes a method for expanding brackets containing two terms raised to a power as a polynomial.^{[1]} More specifically, it describes how to express a bracket of the form (x+a)^{n} as a polynomial with terms b_{k}x^{k}a^{n-k}, for various coefficients b_{k}. When n is natural number, the binomial theorem produces a finite length polynomial, but can also be extended to non-integer n. For n not being a natural number, the binomial expansion of (x+a)^{n} is:^{[1]}

Here, is the binomial coefficient and pronounced "n choose k". It is derived from combinatorics and equal to:

These coefficients for constant n form the rows of Pascal's triangle. The name comes from the Latin, bi-nomin, meaning two names (terms).^{[2]}

## Proof

The binomial theorem for integer n can easily be proved using proof by induction.^{[3]} Supposing the theorem is true for n=N. Then:

Multiplying out the bracket, and substituting j=k+1 gives:

Separating the k=0 and j=N+1 terms, the sums can be combined as:

Using a combinations identity, that ,^{[3]} this can be rewritten as:

Hence if the theorem holds for n=N, it must hold for n=N+1. As it is true for n=0, it is true for all natural numbers.

## Generalisation to Non-Integers

When the binomial theorem is extended to non-integer n or n<0, the series it creates is infinite:^{[1]}

As it is an infinite series, it is often referred to as the "binomial series". The binomial coefficients are generalised so that n need not be a natural number as:^{[4]}

for k>0 and 1 for k=0. As this series contains an infinite number of terms, one must worry whether it converges or not. In the case of the binomial expansion, it always converges if |x/a|<1 or if n a natural number.^{[1]}

Other generalisations exist such as the multinomial expansion for brackets with more than two terms and the multi-binomial expansion where several binomial brackets are multiplied together.

## Example

As an example, consider expanding (x+y)^{4}. It would take a while doing it by hand. Using the binomial theorem, we quickly derive:

These coefficients form the 5th row of Pascal's triangle. As an example with negative n, consider (x+1)^{-1}:

which will converge for |x|<1.

## References

- ↑
^{1.0}^{1.1}^{1.2}^{1.3}Binomial theorem from mathworld.wolfram.com - ↑ Etymology from Merriam-Webster.com
- ↑
^{3.0}^{3.1}K.F. Riley, M.P. Hobson, S.J. Bence,*Mathematical Methods for Physics and Engineering*, Cambridge University Press, 3^{rd}ed., 2006 - ↑ Binomial series from tutorial.math.lamar.edu