Counting number of roots inside a circle, using Rouche's theorem,

Question

Using the Argument Principle and applying Rouche's Theorem, I know that there are 6 zeroes of the polynomial $$z^{10}-6z^6+3z^4-1$$ inside the unit circle $|z|=1$, no zeroes inside $|z|=1/2$, but I'm not sure how to compute the number of zeroes inside the circle $|z|=3/2$.

The best I can do, so far, is to compute the number of zeroes inside the larger circle $|z|=2$. I get $10$ zeroes.

So, I know this polynomial must have $10-6 = 4$ zeroes ...between $|z|=1$ and |z|=2.

Thanks in advance,

Answer 1

Hint. Consider $f(x)$ where $x$ is real, $1\le x\le2$. Also $f(iy)$ where $1\le y\le2$.