Question

Find the term independent of $x$ in the expansion of
${(3x-\frac{2}{x^2})}^{15}$

Answer 1

Let the term independent of $x$ in the expansion of ${(3x-\frac{2}{x^2})}^{15}{=}^{10}{c}_5(3x{)}^{15-k}{\left(\frac{-2}{x^2}\right)}^k$

${=}^{10}{c}_5{3}^{15-k}\times {\left(-2\right)}^k\times x^{n-k-2k}.$

For being the term independent of $x$

$15-3k=0$

$3k=15$

$k=5$

Hence, the term independent of $x$ in the expansion of ${(3x-\frac{2}{x^2})}^{15}{=}^{10}{c}_5(3x{)}^{10}{\left(\frac{-2}{x^2}\right)}^5$

${=}^{10}{c}_5{3}^{10}\times {\left(-2\right)}^5.$