The Binomial Theorem

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Expanding brackets such as (3+2x)^5 is tedious by expanding the brackets one by one. Fortunately there is a quicker method using the binomial theorem. It states,

where n is an integer greater than or equal to one.

Example: Expandthen

which simplifies to

and further to

Example: Expandthen

This simplifies to

We may be asked to solve questions involving the coefficients. For example, the coefficient of in the binomial expansion ofis equal to 3 times the coefficient ofFind 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:

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