## The Binomial Expansion

The binomial Theorem allows us to expand many brackets without multiplying each bracket out one by one. It states:

To expandwe could expandwhich would be a very long winded process. Or we could just substitute forandinto the expression for the binomial expansion. Example: Expandthen

which simplifies to

and further to

We may be asked to solve questions involving the coefficients. For example, the coefficient ofin the binomial expansion ofis equal to 3 times the coefficient of .Find n.

Using the binomial expansion, the coefficient ofis

Using the binomial expansion, the coefficient ofis

Hence we can write down the equation

Now we have to perform some trickery: