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if and are complex cube root of unity then find the value o

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if $α an d β$ are complex cube root of unity then find the value of $α^{2} + β^{2} + α β$

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α an d β

are complex cube root of unity then find the value of

α^{2} + β^{2} + α β

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Text solutionVerified

Complex cube roots are

α = \frac{- 1 + i 3}{2}

and

β = \frac{- 1 - i 3}{2}

\Rightarrow α + β = \frac{- 1 + i 3}{2} + \frac{- 1 - i 3}{2} = - 1

\Rightarrow α β = \frac{- 1 + i 3}{2} \times \frac{- 1 - i 3}{2} = \frac{( - 1 ) ^{2} - ( i 3 ) ^{2}}{4} = \frac{1 + 3}{4} = 1

Now,

α^{2} + β^{2} + α β

= (α + β)^{2} - 2 α β + α β

= (α + β)^{2} - α β

= (- 1)^{2} - 1 = 1 - 1 = 0

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Question Text	if $α an d β$ are complex cube root of unity then find the value of $α^{2} + β^{2} + α β$
Answer Type	Text solution:1
Upvotes	150