Answer to Question #117179 in Algebra for Edward

Question

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

Answer 1

$z^4 + z^3 + z − 1 = 0$

$z(z^2+1)+z^4-1=0$

$z(z^2+1)+(z^2-1)(z^2+1)=0$

$(z^2+1)(z+z^2-1)=0$

$z^2+1=0, z+z^2-1=0$

$z^2+1=0$

$z^2=-1$

$z_1=i$ $z_2=-i$

$z+z^2-1=0$

$D=1+4=5$

$z_3=\frac{-1+\sqrt{5}}{2}$ $z_4=\frac{-1-\sqrt{5}}{2}$

Answer

The three other roots are $z_2,z_3,z_4$ .