<p>If \[\begin{vmatrix} x-4 & 2x & 2x \\ 2x & x-4 & 2x \\ 2x & 2x & x-4 \end{vmatrix} = (A + Bx)(x - A)^2,\] then the ordered pair \((A, B)\) is equal to</p>
Step-by-Step Solution
Key Concept: Factor the determinant by recognizing it has a cyclic structure. Add all rows to find a common factor (x+2), then use row operations to reduce it to a product form involving (x-A)².
<p><strong>Step 1:</strong> Recognize the cyclic symmetric structure. Add R₁ + R₂ + R₃:</p><p>R₁ + R₂ + R₃ = [5x-4, 5x-4, 5x-4] = (5x-4)[1, 1, 1]</p><p><strong>Step 2:</strong> Factor out (5x-4) by replacing R₁ with R₁ + R₂ + R₃, then factoring:</p><p>det = (5x-4)·∣∣1, 1, 1 | 2x, x-4, 2x | 2x, 2x, x-4∣∣</p><p><strong>Step 3:</strong> Perform row operations on the resulting determinant: R₂ → R₂ - 2xR₁ and R₃ → R₃ - 2xR₁:</p><p>det = (5x-4)·∣∣1, 1, 1 | 0, -4-3x, 0 | 0, 2-4x, -4-3x∣∣</p><p><strong>Step 4:</strong> Expand along C₁ and simplify:</p><p>det = (5x-4)[(-4-3x)(-4-3x) - 0] = (5x-4)(4+3x)²</p><p><strong>Step 5:</strong> Rewrite in the form (A + Bx)(x - A)²:</p><p>(5x-4)(3x+4)² = (5x-4)(3)²(x + 4/3)²</p><p>Matching with (A + Bx)(x - A)²: A = -4, B = 5</p><p>∴ Answer: (A, B) = (-4, 5)</p>
Correct Answer: D