# 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:

Example 1
$(x+y)^3=(x+y)(x+y)(x+y)=(x^2+2xy+y^2)(x+y)=x^3+3x^2y+3xy^2+y^3$
Example 2
$(x+y)^4=(x+y)^3(x+y)=x^4+4x^3y+6x^2y^2+4xy^3+y^4$ using the expansion above.
Expanding $(x+y)^n$ by hand for larger n becomes a tedious task. The Edexcel Formula Booklet provides the following formula for binomial expansion:

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

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.

Example 3 – Find the first three terms, in descending powers of x, in the binomial expansion of $(2x+4)^5$.
This can be done using the formula above. Perform a direct substitution as follows: a=2x, b=4 and n=5 and take the first three terms.

$(2x+4)^5=(2x)^5+\left(\begin{array}{c}5\\1\end{array}\right)(2x)^{5-1}(4)+\left(\begin{array}{c}5\\2\end{array}\right)(2x)^{5-2}(4)^2+...$
Note that
$\left(\begin{array}{c}5\\1\end{array}\right)=\frac{5!}{(5-1)!1!}=\frac{120}{20\times 1}=5$
and
$\left(\begin{array}{c}5\\2\end{array}\right)=\frac{5!}{(5-2)!2!}=\frac{120}{6\times 2}=10$
and so the formula becomes
$(2x+4)^5=(2x)^5+5\times(2x)^{4}(4)+10\times(2x)^{3}(4)^2+...=32x^5+320x^4+1280x^3+...$

Note that 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.