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Question
If `omega` is a complex cube root of unity, show that `(2 - omega)(2 - omega^2)` = 7
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Solution
`omega` is a complex cube root of unity.
∴ `omega^3 = 1 and 1 + omega + omega^2` = 0
Also, `1 + omega^2 = -omega, 1 + omega = - omega^2 and omega + omega^2` = – 1
L.H.S. = `(2 - omega)(2 - omega^2)`
= `4 - 2omega^2 - 2omega + omega^3`
= `4 - 2(omega^2 + omega) + 1`
= 4 – 2(– 1) + 1
= 4 + 2 + 1
= 7
= R.H.S.
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