Question

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

Answer 1

$\begin{array}{c}\left(3-2x\right){\left(1+3x\right)}^{-3}\\ \Rightarrow \left(3-2x\right)\left[1-3\left(3x\right)+\frac{\left(-3\right)\left(-4\right){\left(3x\right)}^2}{2!}+\frac{\left(-3\right)\left(-4\right)\left(-5\right){\left(3x\right)}^3}{3!}+........\right]\\ \Rightarrow \left(3-2x\right)\left[1-9x+54x^2-270x^3+...........\right]\\ \Rightarrow -270\times 3x^3-104x^3\\ \Rightarrow \left(-810-104\right)x^3\\ =-914x^3\end{array}$

Hence, coefficient of $x^3$ is $(-914).$