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

Explanation

\(x - \frac{3}{x^{2}} = x - 3x^{-2}\)

Let the power on x be t, so that the power on \(x^{-2}\) = 9 - t

\((x)^{t}(x^{-2})^{9 - t} = x^{3} \implies t - 18 + 2t = 3\)

\(3t = 3 + 18 = 21 \therefore t = 7\)

To obtain the coefficient of \(x^{3}\), we have

\(^{9}C_{7}(x)^{7}(3x^{-2))^{2} = \frac{9!}{(9 - 7)! 7!}(x)^{7}(9x^{-4})\)

= \(\frac{9 \times 8 \times 7!}{7! 2!} \times 9(x^{3}) = 324x^{3}\)