TITLE: What is meant by "the coefficients of a polynomial function $f(x)$ are symmetric functions of its roots"?
QUESTION [5 upvotes]: I am reading Rotman's Introduction to Group Theory. 
One of his first remarks is that:

By the middle of the eighteenth
  century, it was realized that permutations of the roots of a polynomial $f(x)$
  were important; for example, it was known that the coefficients of $f(x)$ are
  "symmetric functions" of its roots.

Aren't the coefficients of a polynomial function constants? How can they then be functions, let alone "symmetric functions"?

REPLY [4 votes]: But that's the ingenuity that revolutionized the theory of solving polynomial equations - it's sort of the reverse of the usual situation! Normally we're given an actual polynomial and want to find its roots. 
However, that doesn't stop us from deciding that we want certain roots and asking, "What kind of polynomial can I write down with these roots?" (By the way, questions like this arise quite naturally when you're writing quizzes and tests for students of the kind of algebra you learned years ago!)
Let's say we want a quadratic with roots $x = r_1$ and $x = r_2$. In this case, any quadratic of the form
$$p(x) = a(x - r_1)(x - r_2) = ax^2 -a(r_1 + r_2)x + ar_1r_2$$
has the roots we want, $r_1$ and $r_2$. So our coefficients for $x^2,\, x,$ and $1$ are respectively 

$a$ (which is completely independent of both $r_1$ and $r_2$; we could swap them, and it wouldn't change $a$)
$-a(r_1 + r_2)$ which again remains invariant under permuting the two roots, and
$ar_1r_2$, another function of $r_1$ and $r_2$ that doesn't depend on which is which.

This is an example of the sort of thing they mean.