Showing how the roots of this complex polynomial are different.

Question

I want to show that the complex polynomial $p(z) = z^5 + 6z - 1$ has four different roots in the annulus $\{z \in \mathbb{C} : \frac{3}{2} < |z| < 2 \}$.

I used Rouché's theorem to proof that $p(z)$ has exactly four roots in the annulus, but I don't see why they should be different.

Any ideas? Thanks.

Answer 1

If they are not different, they are common roots of $p(z)$ and $p'(z)=5z^4+6$ and also of $5p(z)-zp'(z)=24z-5$ (but of course $\frac5{24}$ is not).

Answer 2

A polynomial $p$ has a repeated root if and only if $p$ and $p'$ have a common divisor (as a polynomial) of degree $\ge 1$. You can find the gcd using the Euclidean algorithm.