Find the term containing $x^{2}$ , if any, in the expansion of $(3 x - \frac{1}{2 x})^{8}$

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Solution

${(3 x - \frac{1}{2 x})}^{8} \Rightarrow {(a + b)}^{n} = \sum_{r = 0}^{n} (\begin{matrix} n r \end{matrix}) a^{n - r} b^{r}$
$T_{r + 1} =^{n} C_{r} a^{n - r} b^{r}$
$T_{r + 1} =^{8} C_{r} {(3 x)}^{8 - r} {(\frac{1}{2 x})}^{r}$
$8 - r - r = 2$
$2 r = 6$
$r = 3$
$T_{4} =^{8} C_{r} {(3 x)}^{5} {(\frac{- 1}{2 x})}^{3}$

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