<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 angle subtraction formulas, then substitute into the polynomial. Notice that cot(π/12) satisfies a special algebraic relation that simplifies the quartic expression.
<p><strong>Step 1:</strong> Find cot(π/12) using angle formulas.</p><p>cot(π/12) = cot(45° - 30°) = (cot 45° cot 30° + 1)/(cot 30° - cot 45°)</p><p>= (1·√3 + 1)/(√3 - 1) = (√3 + 1)/(√3 - 1)</p><p>Rationalize: = (√3 + 1)²/(3 - 1) = (3 + 2√3 + 1)/2 = (4 + 2√3)/2 = <strong>2 + √3</strong></p><p><strong>Step 2:</strong> Let t = 2 + √3. Note that t satisfies: t² - 4t + 1 = 0</p><p>Verification: (2 + √3)² - 4(2 + √3) + 1 = 7 + 4√3 - 8 - 4√3 + 1 = 0 ✓</p><p><strong>Step 3:</strong> From t² = 4t - 1, compute higher powers:</p><p>t³ = t·t² = t(4t - 1) = 4t² - t = 4(4t - 1) - t = 16t - 4 - t = <strong>15t - 4</strong></p><p>t⁴ = t·t³ = t(15t - 4) = 15t² - 4t = 15(4t - 1) - 4t = 60t - 15 - 4t = <strong>56t - 15</strong></p><p><strong>Step 4:</strong> Substitute into f(x) = x⁴ - 8x³ + 18x² - 6x + 1 - 2√3</p><p>f(t) = (56t - 15) - 8(15t - 4) + 18(4t - 1) - 6t + 1 - 2√3</p><p>= 56t - 15 - 120t + 32 + 72t - 18 - 6t + 1 - 2√3</p><p>= (56 - 120 + 72 - 6)t + (-15 + 32 - 18 + 1) - 2√3</p><p>= 2t + 0 - 2√3 = 2(2 + √3) - 2√3 = 4 + 2√3 - 2√3</p><p><strong>∴ Answer: 4</strong></p>
Correct Answer: 4