Number of zeros of $ z^7+4z^4+z^3+1$

Question

How many zeros does $z^7+4z^4+z^3+1$ have in each of the regions |z|<1 and |z|<2?

I know I should use Rouche's Theorem but I can't find a $|f(z)| > |p(z)-f(z)|$ and $f(z)$ have equal number of solutions as $p(z)$.

Can anyone explain the question with details?

Answer 1

Note that if $|z| \ge 2$, then $|z|^7 - (4 |z|^4+|z|^3+1) \ge 55$, hence all of the roots lie inside $|z|<2$.

Note that if $|z|=1$, then $4 |z|^4+|z|^3+1 - |z|^7 \ge 5$.

Hence $z \mapsto 4z^4+z^3+1$ and $z \mapsto z^7+4z^4+z^3+1$ have the same number of zeros inside $|z|=1$.

Note that if $|z|=1$, then $4 |z|^4-(|z|^3+1) \ge 2$.

Hence $z \mapsto 4z^4+z^3+1$ and $z \mapsto 4z^4$ have the same number of zeros inside $|z|=1$ (that is, four).