If $i$ is a root, then $-i$ is also a root? I have the following question:
Prove that i is a roof of the equation $g(z) = 0$, where $g(z) = z^3 - 3z^2 + z - 3$. Find the other roots of this equation.
The answer says without any working out:
Since i is a root, then -i is a root. (The answers are $[z - i], [z + i]$, and $[z-3]$)
Why is this?
I know that I can factor out [z-3] and then continue from there, however the answer doesn't do that. I also plotted a graph:


However, that doesn't show the roots at i and -i.
Basically, can I assume that for any polynomial with i as a root, -i is also a root? And why?
Thank you.

Any polynomial with real coefficients with root $r = a+ib$ also has root $\bar{r} = a-ib$. This is known as the conjugate root theorem.
