Question

If $z$ be a complex number satisfying $z^4+z^3+2z^2+z+1=0$, then find the value of $|\overrightarrow{z}|$.

Answer 1

Let's factorize.

>z4+z3+2z2+z+1=0>z4+z3+2z2+z+1=0

>z4+z2+z3+z+z2+1=0>z4+z2+z3+z+z2+1=0

>z2(z2+1)+z(z2+1)+1(z2+1)=0>z2(z2+1)+z(z2+1)+1(z2+1)=0

>(z2+1)(z2+z+1)=0>(z2+1)(z2+z+1)=0

>z2+1=0>z2+1=0 or z2+z+1=0z2+z+1=0

z=±−1−−−√z=±−1 or z=−1±1−4√2z=−1±1−42

z=±iz=±i or z=−1±i3√2z=−1±i32

Therefore, zz has the following values

i,−i,−1+i3√2,−1−i3√2i,−i,−1+i32,−1−i32

The value of z′z′ would be

−i,i,−1−i3√2,−1+i3√2−i,i,−1−i32,−1+i32

The value of |z′||z′| would be only +1.