We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive pseudonymised information about your use of our website. cookie policy and privacy policy.
 
+0  
 
0
81
1
avatar

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
 #1
avatar+5220 
+1

\(z=0 \text{ is an obvious solution}\\ |z|=0\)

 

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

 

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

 

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

.
 Apr 15, 2019

17 Online Users

avatar
avatar
avatar