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Question
If α and β are the complex cube root of unity, show that α2 + β2 + αβ = 0
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Solution
α and β are the complex cube roots of unity.
∴ α = `(-1 + "i"sqrt(3))/2` and β = `(-1 - "i"sqrt(3))/2`
∴ α + β = `(-1 + "i"sqrt(3))/2 + (-1 - "i" sqrt(3))/2`
= `(-1 + "i"sqrt(3) - 1 - "i"sqrt(3))/2`
= `(-2)/2`
= – 1
and αβ = `((-1 + "i"sqrt(3))/2)((-1 - "i"sqrt(3))/2)`
= `((-1)^2 - ("i"sqrt(3))^2)/4`
= `(1 - 3"i"^2)/4`
= `(1 + 3)/4`
= 1 ..........[∵ i2 = – 1]
α2 + β2 + aβ = (α + β)2 – αβ
= (– 1)2 – 1
= 1 – 1
= 0
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