Discussion

The coefficient of \(\frac{1}{x^{2}}\) in the binomial expansion of \(\left(3x - \frac{1}{3x}\right)^{4}\) , is

A.\(\frac{4}{7}\)
B.\(\frac{3}{8}\)
C.\(\frac{2}{9}\)
D.\(\frac{4}{9}\)
E.\(\frac{-4}{9}\)

Answer: E

The coefficient is \(\frac{-4}{9}\).

The general term of \(\left(3x - \frac{1}{3x}\right)^{4}\) is

\[ \begin{aligned} T_{k+1} &= {}^{4}C_{k}(3x)^{4-k}\left(-\tfrac{1}{3x}\right)^{k} \\ &= {}^{4}C_{k}(-1)^{k}\,3^{4-2k}\,x^{4-2k} \end{aligned} \]

For \(x^{-2}\) we need \(4 - 2k = -2\), so \(k = 3\):

\[ {}^{4}C_{3}(-1)^{3}3^{-2} = -\frac{4}{9} \]

The option \(\frac{4}{9}\) drops the sign from \((-1)^{3}\); the others use the wrong term or power of 3.

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