# Product of roots of a complex polynomial

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.

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.

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 $$1$$.