<p>The sum of the real values of \(x\) for which the middle term in the binomial expansion of \(\left(\frac{x^3}{3} + \frac{3}{x}\right)^8\) equals 5670 is</p>
Step-by-Step Solution
Key Concept: For even n in binomial expansion, the middle term is the (n/2 + 1)th term. The coefficient of the middle term must equal the given value to find x.
<p><strong>Solution:</strong> In the expansion of $\left(\frac{x^3}{3} + \frac{3}{x}\right)^8$, the middle term is $T_{4+1} = T_5$ (since $n = 8$ is even, therefore middle term is $\frac{n}{2}+1$ th term).</p><p>The middle term coefficient equals 5670, which gives us $\binom{8}{4}$ multiplied by appropriate powers of $\frac{x^3}{3}$ and $\frac{3}{x}$.</p><p>Solving the resulting equation yields $x$ values that sum to 0.</p><p>∴ Answer is (B) 0.</p>
Correct Answer: B