Problem 8 · Polynomial reduction and interpolation · 5 points
Made by AI and not reviewed by a real person.
Let a, b, and c be the roots of x³ − 3x + 1 = 0. What is a⁸/(3a² − 3) + b⁸/(3b² − 3) + c⁸/(3c² − 3)?
- -27
- 28
- 186
- 6
- 0
Show a hint
- Think about how the cubic factors at each root: its other two roots give the denominator a useful meaning. The root equation can also tame the eighth powers.
- Reduce x⁸ modulo the cubic to a quadratic. Then write that quadratic using its values at a, b, and c, and compare coefficients of x².
Show the answer and solution
Answer: B — 28
Write f(x) = (x − a)(x − b)(x − c). At any root r, f′(r) = 3r² − 3 is the product of r’s differences from the other two roots. Using r³ = 3r − 1 repeatedly gives r⁸ = 28r² − 27r + 6. Let R(x) = 28x² − 27x + 6. Interpolating R from its values at a, b, and c, the coefficient of x² is R(a)/f′(a) + R(b)/f′(b) + R(c)/f′(c). The requested sum is therefore 28. The tempting 186 is the sum a⁸ + b⁸ + c⁸; it ignores the denominators.