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Let z be a complex number such that z^5 = 1 and z does not equal 1. Compute z + 1/z + z^2 + 1/z^2

 May 9, 2022
 #1
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Solve for z: z (approx) = 0.618017 + 0.28182i

Then z + 1/z + z^2 + 1/z^2 = 0

 May 9, 2022
 #2
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\(z^5 = 1\\ z^5 - 1 = 0\\ (z - 1)(z^4 + z^3 + z^2 + z + 1) = 0\\ z^4 + z^3 + z^2 + z + 1 = 0 \text{ ($\because z \neq 1$)}\\ \dfrac{z^4 + z^3 + z^2 + z + 1}{z^2} = 0\\ z + \dfrac1z + z^2 + \dfrac1{z^2} = \boxed{-1}\)

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 May 9, 2022

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