What is the number of zeros of the this polynomial?

Question

I wish to know the number of zeros of the polynomial $z^{10}-6z^7+3z^3+1$ in $|z|<1$.

Does it have something to do with Rouche's theorem?

Answer 1

Take $f=-6z^7$ and $g=z^{10}+3z^3+1$, and $K=\{z\in\mathbb{C}:\vert z\vert <1\}$. In $\partial K$ we have $$\vert g\vert \leq \vert z^{10}+3\vert z^3\vert +1=5<6$$ while $$\vert f\vert =6\vert z^7\vert=6$$ so by the Rouche's theorem we have that $f$ and $f+g$ have the same number of zeros in $K$, so the equation have $7$ zeros in $K$