Find number of zeros of $z^{113}-180z^{23}+115z^{7}-32z^2-3z-10$ in annulus $1≤|z|≤2$

Question

I know I have to use Rouché's theorem but I can't figure out how to choose $f$ and $g$

Answer 1

For $|z|=1$ just take $g(z)=z^{113}$ and $f(z)=-180z^{23}+115z^{7}-32z^2-3z-10$. Clearly $|g(z)|<|f(z)|$, so the function has 1 roots in the unity disk. For $|z|=2$ take $f(z)=z^{113}-180z^{23}+115z^{7}$ and $g(z)=-32z^2-3z-10$. Now clearly $|g(z)|<|f(z)|$ and $g(z)$ has the two zeros in $|z|<2$, hence the original function has $2-1=1$ has two zeros in that region