Question

Find the term independent of x in the expansion of (3x²/2-1/3x)⁹

Answer 1

Ok, let's generalise the expression into the form (a.x² + b/x)^n, then the term designated by the integer k, where k ranges from 0 up to n, is given by

nCk.(a.x²)^(n - k).(b/x)^k

where nCk is the usual binomial coefficient n!/[k! (n - k)!]. For the power of x to be zero, we require that

2.(n - k) - k = 0, or 2.n - 3.k = 0

which is obtained with k = (2/3).n. Since in this case n = 9, the required value of k is 6, and the corresponding term will be

9C6.(a.x²)^3.(b/x)^6 = (9*8*7/3*2*1).(3/2)^3.(x^6) (-1/3)^6.(x^-6) = 7/18

without, as you will notice, any trace of an 'x'.