2026 · Q16
If α and β are the roots of the equation 2x² − 2(n+1)x + (n²+n+1) = 0, then what is (2α³ + 2β³) equal to?
Elementary MathematicsAlgebra
- An³ − 1
- B1 − n³
- C−(n³+1)
- D−(2n³+1)
Explanation
From 2x² − 2(n + 1)x + (n² + n + 1) = 0 we get α + β = n + 1 and αβ = (n² + n + 1)/2. Using α³ + β³ = (α + β)³ − 3αβ(α + β), we have 2α³ + 2β³ = 2(n + 1)³ − 3(n + 1)(n² + n + 1). Factoring (n + 1) gives (n + 1)[2(n + 1)² − 3(n² + n + 1)] = (n + 1)(−n² + n − 1) = −(n + 1)(n² − n + 1) = −(n³ + 1), option C.