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avatar+1110 

Let z be a complex number such that \(z + \frac{1}{z} = 1\). Find z^3.

 

I've already tried expanding z. I've also tried just getting z onto one side, but I ended up getting \(z=z-z^2 \rightarrow 0=-z^2\)

 Apr 14, 2019
 #1
avatar+100571 
+1

z +  1/z  = 1

z^2 + 1  = z

z^2 - z + 1   = 0

 

Complete the square on z

 

z^2  - z + 1/4  =   -1 + 1/4

(z - 1/2)^2  = -3/4

z - 1/2  =  ± i √3 / 2

z =  1/2 +   (√3/2) i        or    z   = 1/2  -  ( √3/2) i

 

In each case

 

z^3  =  - 1

 

 

cool cool cool

 Apr 14, 2019

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