Question

Find the term independent of  in the expansion of

     i.       

 

   ii.       

Solution

i.
 

Expression for the general term in the Binomial expansion of  is:

In the given case :

Hence;

Since we are looking for the coefficient of the term independent of  i.e. , so we can  equate

Hence, substituting ;

Becomes;

Hence coefficient of the term independent of  i.e.  is .

ii.
 

It is evident that to get the terms independent of  in the product of , we need the terms in the expansion of which are independent of  and contain .

We have found in (i) that term in the expansion of which is independent of  is .

Hence, we need a term in the expansion of  with .

Expression for the general term in the Binomial expansion of  is:

In the given case :

Hence;

Since we are looking for the coefficient of the term independent of  i.e. , so we can  equate 

Hence, substituting ;

Becomes;

Hence coefficient of the term containing  i.e. is .

Therefore,

Retaining terms independent of ;