Roots of real polynomial $f(x)$ is a real polynomial.
Show that $z=a+bi$ and $\bar z=a-bi$ have the same algebric multiplicity.
I know that if $z=a+bi$ is a root of $f$ then $\bar z=a-bi$ is too, but don't know how to use it...
Thanks

Suppose that $z=a+bi$ and $z=a-bi$ have different algebraic multiplicities, say $m$ and $n$ respectively. Without loss of generality, suppose $m>n$. Then in the complex conjugate of the polynomial $f(x)$, written $\overline{f(x)}$, the algebraic multiplicities of $z=a+bi$ and $z=a-bi$ will be reversed. But since for real polynomials $f(x)=\overline{f(x)}$, this would require that both $m>n$ and $m<n$. Contradiction.
