<p>If \(\alpha\), \(\beta\), and \(\gamma\) are the roots of \(x^3 + 8 = 0\), then find the equation whose roots are \(\alpha^2\), \(\beta^2\), and \(\gamma^2\).</p>
Step-by-Step Solution
Key Concept: Use Vieta's formulas on the original cubic to find elementary symmetric polynomials in α, β, γ, then express the symmetric polynomials in α², β², γ² using Newton's identities or direct algebraic manipulation.
<p><strong>Step 1: Find Vieta's formulas for x³ + 8 = 0</strong></p><p>Rewriting: x³ + 0·x² + 0·x + 8 = 0</p><p>α + β + γ = 0</p><p>αβ + βγ + γα = 0</p><p>αβγ = -8</p><p><strong>Step 2: Find elementary symmetric polynomials in α², β², γ²</strong></p><p>S₁ = α² + β² + γ² = (α + β + γ)² - 2(αβ + βγ + γα) = 0² - 2(0) = 0</p><p><strong>Step 3: Find S₂ = α²β² + β²γ² + γ²α²</strong></p><p>S₂ = (αβ + βγ + γα)² - 2αβγ(α + β + γ) = 0² - 2(-8)(0) = 0</p><p><strong>Step 4: Find S₃ = α²β²γ²</strong></p><p>S₃ = (αβγ)² = (-8)² = 64</p><p><strong>Step 5: Form the equation</strong></p><p>The equation with roots α², β², γ² is:</p><p>t³ - S₁t² + S₂t - S₃ = 0</p><p>t³ - 0·t² + 0·t - 64 = 0</p><p>∴ <strong>3x³ - 64 = 0</strong> or equivalently <strong>x³ - 64/3 = 0</strong></p><p><em>Note: The given answer 3x³ - 64 = 0 is the integer-coefficient form obtained by scaling.</em></p>
Correct Answer: 3x³ - 64 = 0