<p>Let \(f(x) = (x-1)(x-2)(x-3)(c-x) + x^4 - x\). If the coefficient of \(x^3\) in \(f(x)\) is 1, find \(c\).</p>
Step-by-Step Solution
Key Concept: Expand (x-1)(x-2)(x-3)(c-x) strategically by grouping pairs to identify the x³ coefficient, then use the constraint that the total x³ coefficient equals 1.
<p><strong>Step 1:</strong> Expand (x-1)(x-2)(x-3)(c-x) by grouping strategically.</p><p>[(x-1)(c-x)][(x-2)(x-3)] = [cx - x² - c + x][x² - 5x + 6]</p><p>= [−x² + (c+1)x − c][x² − 5x + 6]</p><p><strong>Step 2:</strong> Find the x³ coefficient by multiplying terms that give x³:</p><p>From (−x²)(−5x) = 5x³</p><p>From [(c+1)x](x²) = (c+1)x³</p><p>Coefficient of x³ = 5 + (c+1) = c + 6</p><p><strong>Step 3:</strong> Note that x⁴ − x contributes no x³ term.</p><p>Total coefficient of x³ in f(x) = c + 6</p><p><strong>Step 4:</strong> Set equal to given constraint:</p><p>c + 6 = 1</p><p>c = −5</p><p>∴ Answer: <strong>-5</strong></p>
Correct Answer: -5