Question

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

Answer 1

Putting $z=i$ in the following term $z^{4}+z^{3}+z-1$ , we have

$i^{4}+i^{3}+i-1=1-i+i-1=0$

thus $z=i$ is a root of the equation $z^{4}+z^{3}+z-1=0$

Since the equation $z^{4}+z^{3}+z-1=0$ has the root $z=i$ , then by using the division rule, we have that

$z^{4}+z^{3}+z-1=(z-i)(z^{3}+(i+1)z^{2}+(i-1)z-i)$

Since $z=i$ is a root of the equation $z^{4}+z^{3}+z-1=0$ , then $z=-i$ is also a root, thus

$z^{3}+(i+1)z^{2}+(i-1)z-i=(z+i)(z^{2}+z-1)$

the equation $z^{2}+z-1=0$ can be solved by the general law of the quadratic function , the roots of this equation is are $z= (-1+\sqrt5 )/2$ and $z= (-1-\sqrt5 )/2$

the four roots are

$z=i$ , $z=-i$ , $z= (-1+\sqrt5 )/2$ and $z= (-1-\sqrt5 )/2$