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
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:




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.

Note that
\left(\begin{array}{c}5\\1\end{array}\right)=\frac{5!}{(5-1)!1!}=\frac{120}{20\times 1}=5
\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

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.