Question

Find coefficient of $x^9$ in the expansion ${(x^2-\frac{1}{3x})}^9$

Answer 1

${(x^2-\frac{1}{3x})}^9=\left(\frac{{(3x^3-1)}^9}{{\left(3x\right)}^9}\right)$

$(3x^3-1{)}^9=3^9{x}^{27}{-}^9{C}_1\times 3^8{x}^{24}{+}^9{C}_2\times 3^7{x}^{21}{-}^9{C}_3\times 3^6{x}^{18}+...-1$

Dividing above expression by ${\left(3x\right)}^9$, we get

$\left(\frac{{(3x^3-1)}^9}{{\left(3x\right)}^9}\right)=x^{18}-\frac{{}^9{C}_1}{3}{x}^{15}+\frac{{}^9{C}_2}{9}{x}^{12}-\frac{{}^9{C}_3}{27}{x}^9+...-1$

Therefore, coefficient of $x^9$ is $\frac{{}^9{C}_3}{27}$, or $\frac{84}{27}$.