Question

If α and β are complex cube roots of unity, prove that (1 − α)(1 − β) (1 − α²)(1 − β²) = 9.

Answer 1

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