<p>The sum of the real values of \(x\) for which the middle term in the binomial expansion of \(\left(\dfrac{x^3}{3} + \dfrac{3}{x}\right)^8\) equals 5670 is ___________.</p>
Step-by-Step Solution
Key Concept: The middle term in a binomial expansion of (a+b)^8 is the 5th term (T₅). Set up the middle term formula using binomial coefficients and equate it to 5670 to find x, then sum all real solutions.
<p><strong>Step 1:</strong> Identify the middle term. For (a+b)^8, there are 9 terms total, so the middle term is T₅ (the 5th term).</p><p><strong>Step 2:</strong> Apply the general term formula: T_{r+1} = C(8,r) · (x³/3)^(8-r) · (3/x)^r</p><p>For the middle term (r=4): T₅ = C(8,4) · (x³/3)^4 · (3/x)^4</p><p><strong>Step 3:</strong> Simplify T₅:</p><p>T₅ = 70 · (x¹²/81) · (81/x⁴) = 70 · x⁸ = 5670</p><p><strong>Step 4:</strong> Solve for x:</p><p>70x⁸ = 5670</p><p>x⁸ = 81</p><p>x⁸ = 3⁴</p><p>x = ±√3 (real solutions: x = √3 and x = -√3)</p><p><strong>Step 5:</strong> Find the sum of all real values:</p><p>√3 + (-√3) = 0</p><p>∴ Answer: <strong>0</strong></p>
Correct Answer: 0