Product of roots of a complex polynomial

Question

I am confused as to why the product of the roots of $$az^n + z + 1$$ is $$\frac{(-1)^n}{a}$$

Can We generalize to other polynomials? Thank you!

This is in the solution of an exercise that uses Rouché's theorem in complex analysis.

Answer 1

Let the roots of a monic polynomial $p(x)$ of degree $n$ be $\alpha_i, i \in \{1, \ldots, n\}$.

Then we have

$$p(x)=\prod_{i=1}^n(x-\alpha_i)$$

and the constant term is $p(0)=\prod_{i=1}^n(-\alpha_i)=(-1)^n\prod_{i=1}^n\alpha_i $

Try to make your polynomial monic and see the result.

Answer 2

The polynomial factors as

$$a(z-z_1)(z-z_2)\cdots(z-z_n)$$ and the independent term is given by

$$P(0)=a(-z_1)(-z_2)\cdots(-z_n)$$

which is know to be $1$.