<p>The constant term in the expansion of \(\left(x^2 - \dfrac{3}{x}\right)^6\) is:</p>
Step-by-Step Solution
Key Concept: For the constant term in a binomial expansion, the powers of the variable must cancel out. In the general term of (x² - 3/x)⁶, set the total power of x to zero to find which term is constant.
<p><strong>Step 1:</strong> Write the general term of (x² - 3/x)⁶</p><p>T_{r+1} = C(6,r) · (x²)^(6-r) · (-3/x)^r = C(6,r) · (-3)^r · x^(12-2r) · x^(-r)</p><p><strong>Step 2:</strong> Combine powers of x</p><p>T_{r+1} = C(6,r) · (-3)^r · x^(12-2r-r) = C(6,r) · (-3)^r · x^(12-3r)</p><p><strong>Step 3:</strong> For constant term, set exponent of x to zero</p><p>12 - 3r = 0 ⟹ r = 4</p><p><strong>Step 4:</strong> Calculate the constant term</p><p>T₅ = C(6,4) · (-3)⁴ = 15 · 81 = 1215 = 5 · 3⁵</p><p>∴ Answer: <strong>5 · 3⁵</strong> or <strong>1215</strong></p>
Correct Answer: 5 · 3^5