Question

Find the value of *a* for which there is no term independent of *x* in the expansion of `(1 + ax^2)(2/x - 3x)^6`?

Answer 1

a = 2

Explanation:

`(1 + ax^2)(2/x - 3x)`
`= (1 + ax^2)(729x^6 + 64/x^6 - 2916x^4 - 576/x^4 + 4860x^2 + 2160/x^2 -4320)`

Upon expansion, the constant term must be eliminated to ensure complete dependence of the polynomial on x. Notice that the `2160/x^2` term becomes `2160a + 2160/x^2` upon expansion.
Setting a = 2 eliminates the constant as well as `2160a`, which was independent of x. (`4320 - 4320)`

(Correct me if I'm wrong, please)