<p>Find the coefficient of \(x^3\) in the expansion of \((1 + x + 2x^2)\left(2x^2 - \dfrac{1}{3x}\right)^9\).</p>
Step-by-Step Solution
Key Concept: The coefficient of x³ comes from multiplying terms in (1 + x + 2x²) with terms from the binomial expansion of (2x² - 1/3x)⁹ such that powers combine to give x³. Specifically, identify which terms from (2x² - 1/3x)⁹ have degrees that when multiplied by 1, x, or 2x² yield x³.
<p><strong>Step 1:</strong> Expand (2x² - 1/3x)⁹ using binomial theorem. The general term is:</p><p>T_{r+1} = C(9,r)(2x²)^{9-r}(-1/3x)^r = C(9,r)·2^{9-r}·(-1/3)^r·x^{18-2r-r} = C(9,r)·2^{9-r}·(-1/3)^r·x^{18-3r}</p><p><strong>Step 2:</strong> Multiply (1 + x + 2x²) with the expansion. To get x³, we need:</p><p>(1) × [coefficient of x³]: Set 18-3r = 3, so r = 5</p><p>Coefficient = C(9,5)·2⁴·(-1/3)⁵ = 126·16·(-1/243) = -2016/243</p><p><strong>Step 3:</strong> (x) × [coefficient of x²]: Set 18-3r = 2, so r = 16/3 (not integer, no contribution)</p><p><strong>Step 4:</strong> (2x²) × [coefficient of x¹]: Set 18-3r = 1, so r = 17/3 (not integer, no contribution)</p><p><strong>Step 5:</strong> Only one valid case exists. Simplify -2016/243:</p><p>-2016/243 = -224/27</p><p>∴ Answer: <strong>-224/27</strong></p>
Correct Answer: -224/27