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Let alpha, beta, gamma be the roots of x³ - 3x² + 3x + 7 = 0, and let w be a non-real cube root of unity. Find the value of (alpha-1)/(beta-1) + (beta-1)/(gamma-1) + (gamma-1)/(alpha-1).

1. 3w²
2. 3w
3. 3
4. 0

Correct answer: 3w²

Solution

Since x³ - 3x² + 3x + 7 = (x-1)³ + 8 = 0, the values (alpha-1), (beta-1), (gamma-1) are -2, -2w, -2w²; each ratio in the sum equals w², giving 3w².

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