Binomial theorem

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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]


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.


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.


  1. 1.0 1.1 1.2 1.3 Binomial theorem from
  2. Etymology from
  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

See also