Question

Find the coefficient of x^-7 in the expansion of ( 2x-1/3x^2)

Answer 1

By the Binomial Theorem, [2x-(1/2x)]^6 can be expanded to the sum as k goes from zero to 6 of (6 choose k)×(2x)k×(-1/2x)6-k.   The only time we will be left with an x2 term is when k=4, since we'll have some coefficient multiplied by x4/x2.  So, we'll have (6 choose 4) × (2x)4×(-1/2x)2, which is 15×16x4×(1/4x2)= 60x2.

 

So the coefficient is 60.