This page details the more advanced use of binomial expansion. You should be familiar with all of the material from the more basic Binomial Expansion page first.
Recall that the first formula provided in the Edexcel formula booklet is:
for and where . However, this formula is only valid for positive integer .
In addition to this, the booklet also provides a second formula for negative and fractional powers:
The first formula is only valid for positive integer but this formula is valid for all . This includes negative and fractional powers. Note, however, the formula is not valid for all values of . As stated, the values must be between -1 and 1.
Range of Validity for Binomial Expansions
As stated above, the second formula for binomial expansion in the Edexcel Formula Booklet is only valid for . This is because, unlike for positive integer , these expansions have an infinite number of terms (as indicted by the … in the formula). Subsequently, we require the series to converge as the powers of become large. For this to happen, we must have . Questions may ask you to find the binomial expansion without explicitly stating the value of and ask you to identify the values for which the expansion is valid. See the Identifying the Power Example.
Also notice that in this second formula there is a very specific format inside the brackets – it must be 1 plus something. Therefore, if there is something other than 1 inside these brackets, the coefficient must be factored out. Do this by first writing . Then find the expansion of using the formula. Do this by replacing all with . This inevitably changes the range of validity. It follows that this expansion will be valid for or . See the Factoring Out Example.
At more advanced levels, questions may ask you to use partial fractions first. See the Using Partial Fractions question.