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Let $z$ be a complex number such that z^5 + z^4 + 2z^3 + z^2 + z = 0. Find all possible values of |z|. 

 Apr 14, 2019
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\(z=0 \text{ is an obvious solution}\\ |z|=0\)

 

\(\text{Assume }z\neq 0\\ z^5 + z^4+2z^3 + z^2 + z = 0\\ z^4 + z^2 = -z(z^4 + 2z^2+1)\\ z(z^2+1) = -(z^2+1)^2\)

 

\(\text{if }z^2\neq -1\\ z=-(z^2+1)\\ z^2+z+1=0\\ z = \dfrac{-1\pm\sqrt{-3}}{2}=-\dfrac 1 2 \pm i \dfrac{\sqrt{3}}{2}\\ |z| = \sqrt{\dfrac 1 4 + \dfrac 3 4 }=1\)

 

\(\text{if }z^2=-1\\ z=\pm i\\ |z|=1\)

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 Apr 15, 2019

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