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Question
If α and β are complex cube roots of unity, prove that (1 − α)(1 − β) (1 − α2)(1 − β2) = 9.
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Solution
Since α and β are the complex cube roots of unity, α2 = β and β2 = α
Also, α3 = 1, 1 + α + α2 = 0
∴ α4 = α3.α = α, 1 + α2 = – α and 1 + α = – α2
∴ (1 – α)(1 – β)(1 – α2)(1 – β2)
= (1 – α)(1 – α2)(1 – α2)(1 – α)
= (1 – α)2(1 – α2)2
= (1 + α2 – 2α)(1 + α4 – 2α2)
= (1 + α2 – 2α)(1 + α – 2α2) ...[∵ α4 = α]
= (– α – 2α)(– α2 – 2α2)
= (– 3α)(– 3α2)
= 9 α3
= 9 × 1
= 9
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