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Given that the cube roots of unity are 1, ω, and ω², which of the following is the set of roots of the equation (x - 1)³ + 8 = 0?

1. -1, -1 + 2ω, -1 - 2ω²
2. -1, -1, -1
3. -1, 1 - 2ω, 1 - 2ω²
4. -1, 1 + 2ω, 1 + 2ω²

Correct answer: -1, 1 - 2ω, 1 - 2ω²

Solution

The equation can be rewritten as (x - 1)³ = -8, which simplifies to (x - 1) = -2, leading to x = -1. The other roots are derived from the cube roots of -8, which correspond to the expressions 1 - 2ω and 1 - 2ω², making option C the correct choice.

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