Question

Show that z = i is a root of the equation z4 +z3 +z−1 = 0. Find the three other roots.

Answer 1

Equation z⁴ +z³ +z−1 = 0 can be written after re-arranging the terms

=> (z⁴-1) +z(z²+1)=0

=> (z²-1)(z²+1)+z(z²+1)=0

=> (z²+1) (z²+z-1) =0

=> So roots of z² +1 =0 are -i and i;

roots of (z² +z-1) =0 are $(-1+\sqrt{\smash[b]{5}})/2$ and $(-1-\sqrt{\smash[b]{5}})/2$.

hence other 3 roots are  -i, $(-1+\sqrt{\smash[b]{5}})/2$ and $(-1-\sqrt{\smash[b]{5}})/2$ .