<p>In the expansion of \((1 + x)^{2m}\left(\dfrac{x}{1-x}\right)^{-2m}\), the term independent of \(x\) is</p>
Step-by-Step Solution
Key Concept: Rewrite the expression as (1+x)^(2m) · (1-x)^(2m) / x^(2m), then find the coefficient of x^(2m) in (1+x)^(2m)(1-x)^(2m) using the identity (1-x²)^(2m). The constant term emerges when powers of x cancel perfectly.
<p><strong>Step 1:</strong> Rewrite the given expression:</p><p>(1 + x)^(2m) · (x/(1-x))^(-2m) = (1 + x)^(2m) · (1-x)^(2m) / x^(2m)</p><p><strong>Step 2:</strong> Simplify the numerator using algebraic identity:</p><p>(1 + x)^(2m) · (1-x)^(2m) = [(1+x)(1-x)]^(2m) = (1-x²)^(2m)</p><p><strong>Step 3:</strong> The expression becomes:</p><p>(1-x²)^(2m) / x^(2m)</p><p><strong>Step 4:</strong> Expand (1-x²)^(2m) using binomial theorem:</p><p>(1-x²)^(2m) = Σ C(2m,r)(-x²)^r = Σ C(2m,r)(-1)^r · x^(2r)</p><p><strong>Step 5:</strong> The general term in the full expression is:</p><p>C(2m,r)(-1)^r · x^(2r) / x^(2m) = C(2m,r)(-1)^r · x^(2r-2m)</p><p><strong>Step 6:</strong> For the term independent of x, set 2r - 2m = 0, giving r = m</p><p><strong>Step 7:</strong> The constant term is:</p><p>C(2m,m)(-1)^m = (-1)^m · (2m)!/(m!)²</p><p>∴ Answer: <strong>(-1)^m · C(2m,m)</strong></p>
Correct Answer: A