State true or false:
The roots of equation ${(x - 1)}^{3} + 8 = 0$ are $- 1, 1 - 2 ω, 1 - 2 ω^{2} .$

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Solution

${(x - 1)}^{3} + 8 = 0$ or ${(x - 1)}^{3} = - 8.$
$∴ x - 1 = {(- 8)}^{1 / 3} = - 2, - 2 ω, - 2 ω^{2} .$
Hence $x = - 1, 1 - 2 ω, 1 - 2 ω^{2} .$

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