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if Z is a complex no. satisfying z^4+z^3+2z^2+z+1=0, then find value of modulus of z?

if Z is a complex no. satisfying z^4+z^3+2z^2+z+1=0, then find value of modulus of z?


 


 


 


 


 


 


 


 


 


 


 


 


 


 


 


 


 


 

Grade:

2 Answers

Ashwin Muralidharan IIT Madras
290 Points
9 years ago

Hi Vishnu Priya,

 

Please note, the given expression is

z4+z3+z2+z2+z+1 = z2(z2+z+1)+1(z2+z+1) = (z2+1)(z2+z+1)

So that is equal to 0, will mean either z2 = -1 or z2+z+1=0

So either z=i or z=w,w2 (w is the cube root of unity)

 

So in all cases |z|=1.

 

Hope that helps.

 

All the best.

Regards,

Ashwin (IIT Madras).

Siddhant Somani
29 Points
9 years ago

z4+z3+z2+z2+z+1 = z2(z2+z+1)+1(z2+z+1) = (z2+1)(z2+z+1)

Therefore z2 = -1 or z2+z+1=0

So either z=i or z=w,w2

Hence..

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