Question

Find the term independent of $x$ in the expansion of $(1+x+2x^2){(\frac{3x^2}{2}-\frac{1}{3x})}^9.$

Answer 1

General term is $T_{r+1}{=}^9{C}_r{\left(\frac{3x^2}{2}\right)}^{9-r}{\left(\frac{-1}{3x}\right)}^r(1+x+2x^2)$

This term to be independant consider powers of $x$

$x^{18-2r}{x}^{-r}=1⟹18-3r=0⟹r=6$

so the term is $T_7{=}^9{C}_6{\left(\frac{3x^2}{2}\right)}^3{\left(\frac{-1}{3x}\right)}^6⟹84\times \frac{27}{8}\times \frac{1}{729}⟹\frac{7}{18}$