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. More specifically, it describes how to express a bracket of the form (x+a)ⁿ as a polynomial with terms b_kx^ka^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:

$(x+a)^{n}=\sum _{k=0}^{n}{n \choose k}x^{k}a^{n-k}$

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

${n \choose k}={\frac {n!}{(n-k)!k!}}$

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

Proof

The binomial theorem for integer n can easily be proved using proof by induction. Supposing the theorem is true for n=N. Then:

$(x+a)^{N+1}=(x+a)\sum _{k=0}^{N}{N \choose k}x^{k}a^{N-k}$

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

$(x+a)^{N+1}=\sum _{k=0}^{N}{N \choose k}x^{k}a^{N+1-k}+\sum _{j=1}^{N+1}{N \choose j-1}x^{j}a^{N+1-k}$

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

$(x+a)^{N+1}=a^{(N+1)}+\sum _{k=1}^{N}\left({N \choose k}+{N \choose k-1}\right)x^{k}y^{(N+1)-k}+{N+1 \choose N+1}x^{N+1}$

Using a combinations identity, that ${N \choose k}+{N \choose k-1}={N+1 \choose k}$, this can be rewritten as:

$(x+a)^{N+1}=\sum _{k=0}^{N+1}{N+1 \choose k}x^{k}a^{N-k}$

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:

$(x+a)^{n}=\sum _{k=0}^{\infty }{n \choose k}x^{k}a^{n-k}=a^{n}+nxa^{n-1}+{\frac {n(n-1)}{2!}}x^{2}a^{n-2}+{\frac {n(n-1)(n-2)}{3!}}x^{3}a^{n-3}...$

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:

${n \choose k}={\frac {1}{k!}}\prod _{i=1}^{k}(n-k+1)={\frac {n(n-1)(n-2)...(n-k+1)}{k!}}$

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.

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:

$(x+y)^{4}=x^{4}+4x^{3}y+6x^{2}y^{2}+4xy^{3}+y^{4}$

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

$(x+1)^{-1}=1-x+{\frac {1}{2}}x^{2}-{\frac {1}{6}}x^{3}+{\frac {1}{24}}x^{4}-...$

which will converge for |x|<1.

See also

• Polynomial
• Combinatorics
• Pascal's triangle