Understanding a sentence in a Galois Theory paper about polynomials I've been trying to understand a sentence of a math for some time. I'll be straightforward:
(*) $$x^n + a_{n-1}x^{n-1} +...+a_1x + a_0 = 0$$
(...)
If we denote the roots of (*) by $x_1, x_2, ... , x_n$ so that
$$(x-x_1)(x-x_2)...(x-x_n) = x^n + a_{n-1}x^{n-1} +...+a_1x + a_0$$

Then $a_0, ... , a_{n-1}$ are polynomial functions of $x_1, x_2, ... , x_n$ called elementary symmetric functions:
$$a_0 = (-1)^nx_1x_2...x_n , a_{n-1} = -(x_1 + x_2 + ... + x_n)$$

This last statement between the two lines is the one I have some issues with. 


*

*When the author claims that $a_0, ... , a_n$ are functions of $x_1, ... , x_n$, does he simply mean it in a way that $x_1, ... , x_n$ define what the elements $a_0, ... , a_n$ are?

*And more importantly, where does this last equation come from? The only way I can think of making a_0 a function of $x_1, ... , x_n$ and $a_1, ... , a_n$ is to use (*) to write 
$$a_0 = (-1)[x^n + a_{n-1}x^{n-1} +...+a_1x]$$
, but I can't seem to relate this last equality to the one the author wrote.

I would truly appreciate any help/thoughts!

For $1$, what darij said. 
Just expand $(x-x_1)(x-x_2)\cdots (x-x_n)$. For small $n$, we have:
$$(x-x_1)(x-x_2) = x^2 - (x_1 + x_2) x + x_1 x_2 \\(x-x_1)(x-x_2)(x-x_3) = x^3 - (x_1 + x_2 + x_3)x^2 + (x_1 x_2 + x_1 x_3 + x_2 x_3)x - x_1 x_2 x_3$$
and the pattern reveals itself as the author claims: 
$$(x-x_1)\cdots(x-x_n) = x^n - (x_1 + \cdots + x_n) x^{n-1} + \cdots + (-1)^n x_1 \cdots x_n$$
now compare these two expressions of $x^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0$.
