Question

2. (a) Find the real root of the equation z3 + z + 10 = 0 given that one root is 1 − 2i.
Show that z = i is a root of the equation z4 + z3 + z − 1 = 0. Find the three
other roots.

Answer 1

$z=1+2i$ is also a root of equation. We obtain this by conjugating the equation. Dividing the polynomial $z^3+z+10$ by $(z-1-2i)(z-1+2i)=z^2-2z+5$

we receive $z+2$ . Thus, $z=-2$ is the real root.

Now we consider equation $z^4+z^3+z-1=0$ . We will check that $z_1=i$ is a root of equation. Namely, we have: $z_1^4+z_1^3+z_1-1=1-i+i-1=0$

It is clear that $z_2=-i$ is also a root of equation.

This can be obtained from the fact that $z_2=\bar{z}_1$

We shall divide the equation by $(z-i)(z+i)=z^2+1$ .

Then one receives $z^2-1+z.$ It remains to solve $z^2+z-1=0$ .

The latter quadratic equation has solutions (see e.g. https://en.wikipedia.org/wiki/Quadratic_equation for detalis)

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

Thus, we have the roots $z_1=i,\quad z_2=-i,\quad z_3 = \frac{-1+\sqrt{5}}2,\quad z_4 = \frac{-1-\sqrt{5}}2,$