Find the real root of the equation $z^3 + z + 10 = 0$ given that one complex root is $1 – 2i$.
I've realized that the roots are $(1-2i), (1+2i)$, and a real number we'll call $a$.
So using the theorem got me $(z-1-2i)(z-1+2i)(z-x)$.
No idea on where to go next.
Best Answer
The polynomial is monic (lead coefficient $1$). The coefficient of $z^2$ is therefore the negative of the sum of the roots. This coefficient is $0$.
The two known roots have sum $2$, so the missing root must be $-2$.