# 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?

Apr 30, 2018

a = 2

#### Explanation:

$\left(1 + a {x}^{2}\right) \left(\frac{2}{x} - 3 x\right)$
$= \left(1 + a {x}^{2}\right) \left(729 {x}^{6} + \frac{64}{x} ^ 6 - 2916 {x}^{4} - \frac{576}{x} ^ 4 + 4860 {x}^{2} + \frac{2160}{x} ^ 2 - 4320\right)$

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

(Correct me if I'm wrong, please)