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Let a, b, c be the roots of the polynomial x³ - 2x² + x - 1 = 0. If S = a⁵ + b⁵ + c⁵, evaluate 4*S / 17.

1. 1
2. 2
3. 4
4. 8

Correct answer: 4

Solution

By Vieta's formulas, the elementary symmetric sums are a+b+c=2, ab+bc+ca=1, abc=1. Newton's identity gives pₙ=2pₙ₋₁-pₙ₋₂+pₙ₋₃. Starting from p₀=3, p₁=2, p₂=2, successive application yields p₃=5, p₄=10, p₅=17. Thus 4*17/17=4.

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