Question

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

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

Hint:

Always equate powers of $x$ in general term to find required coefficient.

Answer 1

Answer: (E)

Concept:
• General term of binomial expansion: \[ T_{k+1} = \binom{n}{k} (a)^{n-k} (b)^k \]
• Match power of $x$ to required term

Step 1: Write general term
\[ T_{k+1} = \binom{4}{k} (3x)^{4-k} \left(-\frac{1}{3x}\right)^k \]

Step 2: Simplify powers of $x$
\[ = \binom{4}{k} \cdot 3^{4-k} x^{4-k} \cdot (-1)^k \cdot \frac{1}{3^k x^k} \] \[ = \binom{4}{k} (-1)^k \cdot 3^{4-2k} \cdot x^{4-2k} \]

Step 3: Find power condition
\[ 4 - 2k = -2 \Rightarrow 2k = 6 \Rightarrow k = 3 \]

Step 4: Substitute $k=3$
\[ T_4 = \binom{4}{3} (-1)^3 \cdot 3^{4-6} \] \[ = 4 \cdot (-1) \cdot 3^{-2} = -4 \cdot \frac{1}{9} \] \[ = -\frac{4}{9} \] Final Conclusion:
Coefficient = $-\frac{4}{9}$