Why is the product of the roots of $p(z) = 1 + z + az^n$ equal to $1$?

Question

This should be very straightforward to show, but I am having some issues doing so. We have:

$$1 + z + az^n = a(z-z_1)(z-z_2)\dots(z-z_n),$$

where $z_1$, $z_2$, $\dots , z_n$ are the roots of $p$. Clearly,

$$1 = (-1)^na(z_1 z_2\dots z_n).$$

I'm pretty much stuck at this point - it seems like the absolute value of the product of the roots will be $1/a$, but this isn't what we want.

I suppose that the problem statement could be wrong - I am working out of the Berkeley book of problems and solutions in mathematics, and this is part of an answer to a problem. But, I'm checking in to be sure.

Answer 1

Consider $1+z+2z^2 $.

The roots are $\frac{-1\pm\sqrt{1-8}}{4} =\frac{-1\pm\sqrt{-7}}{4} $ and their product is $\frac{-1+\sqrt{-7}}{4}\frac{-1-\sqrt{-7}}{4} =\frac{1-(-7)}{16} =\frac12 $ which is not $1$.

Answer 2

This is called Vieta's theorem.

If you don't want to use Vieta's theorem, you can also expand the brackets to get this result.

$a$ has to be non-zero.