<p>Let \(f(x) = x^4 - 8x^3 + 18x^2 - 6x + 1 - 2\sqrt{3}\), then \(f\left(x = \cot\dfrac{\pi}{12}\right)\) is equal to:</p>
Step-by-Step Solution
Key Concept: First find the exact value of cot(π/12) using the cotangent subtraction formula or half-angle identities, then substitute into the polynomial. Recognize that cot(π/12) = 2 + √3, which will simplify the polynomial evaluation significantly.
<p><strong>Step 1:</strong> Find cot(π/12). Since π/12 = 15°, use cot(45° - 30°) = (cot 45° cot 30° + 1)/(cot 30° - cot 45°) = (1·√3 + 1)/(√3 - 1) = (√3 + 1)/(√3 - 1). Rationalize: multiply by (√3 + 1)/(√3 + 1) to get (√3 + 1)²/(3 - 1) = (4 + 2√3)/2 = 2 + √3.</p><p><strong>Step 2:</strong> Let x = 2 + √3. Notice that x - 2 = √3, so (x - 2)² = 3, giving x² - 4x + 4 = 3, thus x² = 4x - 1.</p><p><strong>Step 3:</strong> Compute higher powers: x³ = x·x² = x(4x - 1) = 4x² - x = 4(4x - 1) - x = 16x - 4 - x = 15x - 4. Also x⁴ = x·x³ = x(15x - 4) = 15x² - 4x = 15(4x - 1) - 4x = 60x - 15 - 4x = 56x - 15.</p><p><strong>Step 4:</strong> Substitute into f(x): f(x) = (56x - 15) - 8(15x - 4) + 18(4x - 1) - 6x + 1 - 2√3 = 56x - 15 - 120x + 32 + 72x - 18 - 6x + 1 - 2√3 = (56 - 120 + 72 - 6)x + (-15 + 32 - 18 + 1) - 2√3 = 2x + 0 - 2√3 = 2(2 + √3) - 2√3 = 4 + 2√3 - 2√3 = 4.</p><p>∴ Answer: <strong>4</strong></p>
Correct Answer: 4