<p>Find the value of \(x^4 + 9x^3 + 35x^2 - x + 4\) for \(x = -5 + 2\sqrt{-4}\).</p>
Step-by-Step Solution
Key Concept: Recognize that x = -5 + 2√(-4) = -5 + 4i is a root of a quadratic with real coefficients. Use the minimal polynomial x² + 10x + 41 = 0 to reduce higher powers of x, avoiding direct computation of x⁴.
<p><strong>Step 1:</strong> Simplify x. We have x = −5 + 2√(−4) = −5 + 2(2i) = −5 + 4i</p><p><strong>Step 2:</strong> Find the minimal polynomial. Computing x²: x² = (−5 + 4i)² = 25 − 40i − 16 = 9 − 40i. Also, x² + 10x + 41 = (9 − 40i) + 10(−5 + 4i) + 41 = 9 − 40i − 50 + 40i + 41 = 0. So x² + 10x + 41 = 0, giving us <strong>x² = −10x − 41</strong></p><p><strong>Step 3:</strong> Compute x³ and x⁴. x³ = x·x² = x(−10x − 41) = −10x² − 41x = −10(−10x − 41) − 41x = 100x + 410 − 41x = 59x + 410. Then x⁴ = x·x³ = x(59x + 410) = 59x² + 410x = 59(−10x − 41) + 410x = −590x − 2419 + 410x = −180x − 2419</p><p><strong>Step 4:</strong> Substitute into the original polynomial: x⁴ + 9x³ + 35x² − x + 4 = (−180x − 2419) + 9(59x + 410) + 35(−10x − 41) − x + 4 = −180x − 2419 + 531x + 3690 − 350x − 1435 − x + 4 = (−180 + 531 − 350 − 1)x + (−2419 + 3690 − 1435 + 4) = 0·x + (−160) = <strong>−160</strong></p>
Correct Answer: -160