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

If the root is real, the result is obvious, so we may assume that $b\ne 0$. Let us suppose the result is true for all polynomials of degree $\lt n$. We show it is true for polynomials of degree $n$.
So let $w$ be a non-real root of the polynomial $P(x)$ of degree $n$. Then $\bar{w}$ is a root. It follows that the real polynomial $(x-w)(x-\bar{w})$ divides $P(x)$, say $P(x)=(x-w)(x-\bar{w})Q(x)$. By induction hypothesis, $w$ and $\bar{w}$ have the same multiplicity (possibly $0$) as roots of $Q(x)$. It follows they have the same multiplicity as roots of $P(x)$.   
