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


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


which is know to be $1$.


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