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 bkxkan-k, for various coefficients bk. 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]

## Contents

* 1 Proof
* 2 Generalisation to Non-Integers
* 3 Example
* 4 References
* 5 See also

## Proof[edit]

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[edit]

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[edit]

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[edit]

1. ↑ 1.0 1.1 1.2 1.3 Binomial theorem from mathworld.wolfram.com
2. ↑ Etymology from Merriam-Webster.com
3. ↑ 3.0 3.1 K.F. Riley, M.P. Hobson, S.J. Bence, Mathematical Methods for Physics and Engineering, Cambridge University Press, 3rd ed., 2006
4. ↑ Binomial series from tutorial.math.lamar.edu

## See also[edit]

* Polynomial
* Combinatorics
* Pascal's triangle