Past exam of the mathematics course of the University of Cambridge / 2013 / ib / Paper 4 / 4E (Complex Analysis)

Rouché's theorem states that if $f,g$ are holomorphic functions on a neighbourhood of a simple closed contour and its interior, and $|g|<|f|$ on the contour, then $f$ and $f+g$ have the same number of zeros inside, counted with multiplicity.

On $|z|=1$, use $f=6z^2$: the other terms have absolute value at most $1+3+1=5<6$. Hence there are two zeros inside the unit circle. On $|z|=2$, use $f=3z^7$: its modulus is $384$, whereas the other terms have total modulus at most $256+24+1=281$. Hence there are seven zeros inside the larger circle. The strict inequalities also exclude zeros on either boundary. There are $7-2=5$ roots in the annulus, counted with multiplicity.