# Binomial Expansion

Binomial Expansion refers to expanding an expression that involves two terms added together and raised to a power, i.e. $(x+y)^n$. In the simple case where n is a relatively small integer value, the expression can be expanded one bracket at a time. See Examples 1 and 2.

Expanding $(x+y)^n$ by hand for larger n becomes a tedious task. The Edexcel Formula Booklet provides the following formula for binomial expansion:

$(a+b)^n=a^n+\left(\begin{array}{c}n\\1\end{array}\right)a^{n-1}b+\left(\begin{array}{c}n\\2\end{array}\right)a^{n-2}b^2...$
$\hspace{30pt}+...+\left(\begin{array}{c}n\\r\end{array}\right)a^{n-r}b^r+...+b^n$

where

$\left(\begin{array}{c}n\\r\end{array}\right)=\frac{n!}{(n-r)!r!}$

for when $n\in{\mathbb N}$, i.e for when n is a positive integer. See Example 3 and see Factorial Notation to find out about !

Directly substituting a for x and b for y (whatever they might be), results in finding the expansion. Usually only the first few terms are required. If the question says ascending powers of x, then a and b can be switched over so that the powers of x are increasing instead.

### Example 1

Expand $(x+y)^3.$

### Example 2

Using Example 1 expand $(x+y)^4.$

### Example 3

Find the first three terms, in descending powers of x, in the binomial expansion of $(2x+4)^5$.