# Binomial Expansion – positive integer powers

## Binomial Expansion Notes

Binomial Expansion is essentially multiplying out brackets. A binomial is two terms added together and this is raised to a power, i.e. $(x+y)^n$. Before learning how to perform a Binomial Expansion, one must understand factorial notation and be familiar with Pascal’s triangle. Take the Binomial Expansion course.

Suppose that we wish to expand $(x+y)^n$, i.e. find the Binomial Expansion. In the simple case where n is a relatively small integer value, we expand the expression 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:

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

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.  Directly substituting $x$ in place of $a$ and $y$ in place of $b$ results in finding the expansions for larger $n$. Usually only the first few terms are required. You may substitute other expressions or numbers for $a$ and $b$. Note that if question asks you for descending powers of $x$ so you may need to swap the variables accordingly. Also note that when there are also coefficients inside the brackets, the coefficients in the expansion change dramatically from those given in Pascal’s triangle. Take the Binomial Expansion course. Dr Nat

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