<p>Find the sum of the eighth powers of all roots of the equation \(x^3 - x + 1 = 0\)</p>
Step-by-Step Solution
Key Concept: Use Vieta's formulas to find power sum recurrence relations. Since roots satisfy x³ = x - 1, recursively reduce higher powers using this minimal polynomial relation to compute S₈ = x₁⁸ + x₂⁸ + x₃⁸.
<p><strong>Step 1: Establish Vieta's formulas</strong><br>For x³ - x + 1 = 0 with roots α, β, γ:<br>α + β + γ = 0<br>αβ + βγ + γα = -1<br>αβγ = -1</p><p><strong>Step 2: Use the minimal polynomial relation</strong><br>Since each root satisfies x³ = x - 1, we have the recurrence:<br>Sₙ₊₃ = Sₙ₊₁ - Sₙ, where Sₙ = α^n + β^n + γ^n</p><p><strong>Step 3: Calculate initial power sums</strong><br>S₀ = 3<br>S₁ = 0 (from Vieta's)<br>S₂ = (α + β + γ)² - 2(αβ + βγ + γα) = 0 - 2(-1) = 2<br>S₃ = S₁ - S₀ = 0 - 3 = -3</p><p><strong>Step 4: Apply recurrence relation</strong><br>S₄ = S₂ - S₁ = 2 - 0 = 2<br>S₅ = S₃ - S₂ = -3 - 2 = -5<br>S₆ = S₄ - S₃ = 2 - (-3) = 5<br>S₇ = S₅ - S₄ = -5 - 2 = -7<br>S₈ = S₆ - S₅ = 5 - (-5) = 10</p><p><strong>∴ Answer: B (10)</strong></p>
Correct Answer: B