I'm revising for an exam and this should be easy, but I'm not sure if I should use Rouché's Theorem or try solving it via 'traditional' integration along the logarithmic derivative.
What is the number of roots (counting multiplicity) of the polynomial $p(x):= z^5 + 11z^2 − 4z − 2$ in $B_2(0)$?
Suppose we know that $p(z)$ does not have any zeros on $|z|=2$.
With $q(z):= -11z^2$ and for $|z|=2$ we have:
$|p(z)+q(z)|=|z^5−4z−2|≤|z|^5+4|z|+2=42<44=|11z^2| = |q(z)|$.
Thus, by Rouché's Theorem, the number of zeros times their multiplicities and indices of $p$ and $q$ in $B_2(0)$ is equal.
Because $q(z)$ has a zero of second order in $z=0$ and $Ind(\partial B_2(0), 0)=1$ the number is $2$.