I have an exam tomorrow in linear algebra, and I want to make sure I answered this question correctly.
Let $p \in \mathbb R[x], z \in \mathbb{C}$.
We are given if $Im(z)>0$ then $p(z)\neq0$
Show that it is possible to reduce $p$ to linear terms in $\mathbb R[x]$
My solution:
We know that every polynomial of $n$th degree has $n$ roots in $\mathbb{C}$. However, we know that if $b>0$ then $a+bi$is not a root of $p$.
if $a+bi$ is not a root of $p$ then $a-bi$ is not a root of $p$ either because it is it's conjugate.
That mean's all the roots of $p$ are of the form$a+0i=a \in \mathbb{R}$, so $p$ has $n$ roots in $\mathbb{R}$ and so it is reducible to linear terms in $\mathbb R[x]$.