I have a polynomial $p(z)$ of degree $n-1$ with known roots $z_1, \dots, z_{n-1}$. How I add the monomial term $a z^n$. What are the roots of
$$ p_1(z) = p(z) + \epsilon z^n $$
In terms of the roots of $\epsilon$ ? For very small values $a \ll 1$ the roots original $n-1$ roots should not move very much. Can we estimate to first order in $\epsilon$ ? Where does the $n$-th root come from?
Similar problems:
If you divide $p(z)+\epsilon z^n$ by $z^n$, you get a polynomial $q(1/z)+\epsilon=0$. This has a nonzero linear term because $p(z)$ was order $n-1$. So there is a root of $1/z=O(\epsilon)$, and $z=O(1/\epsilon)$. The $n$th root comes from $\infty$.
The change in the other roots depends on their multiplicity. If a root has multiplicity $m$, then they are change by $O(\epsilon^{1/m})$ which is much greater than $O(\epsilon)$. That is because the new polynomial is roughly $c(z-z_k)^m+\epsilon z_k^n =0$ near $z_k$.