I am a little confused by the following exercise in Dummit and Foote, Exercise 14.6.31.
This is the setup of the exercise (not the statement to be proved):
Consider $f(x) = a_nx^n + \cdots + a_0$ and $g(x) = b_mx^m + \cdots + b_0$ as general polynomials and suppose the roots of $f(x)$ are $x_1,\dotsc,x_n$ and the roots of $g(x)$ are $y_1,\dotsc, y_m$. The coefficients of $f(x)$ are powers of $a_n$ times the elementary symmetric functions in $x_1,x_2,\dotsc,x_n$ and the coefficients of $g(x)$ are powers of $b_m$ times the elementary symmetric functions in $y_1,y_2,\dotsc,y_m.$
Earlier in the section, they give the definition
The general polynomial of degree $n$ is the polynomial $$(x - x_1)(x- x_2)\cdots (x-x_n)$$ whose roots are the indeterminates $x_1,x_2,\dotsc,x_n$.
I see how the general polynomial in this definition has coefficients which are elementary symmetric functions in $x_1,\dotsc,x_n$ (just by expanding the expression). I don't understand exactly what it means to consider $f(x)$ and $g(x)$ as general polynomials, since they don't seem to be general polynomials under this definition. I was guessing it meant to view the $a_i$ and $b_j$ as indeterminates and to view $f$ and $g$ as polynomials in $F(a_0,\dotsc,a_n,b_0,\dotsc,b_m)$ or something like that. But then, wouldn't $\,f$, for example, factor as $a_n(x - x_1)\cdots(x - x_n)$? Why would the coefficients of this polynomial involve powers of $a_n$ times the elementary symmetric functions in $x_1,\dotsc,x_n$ as opposed to just $a_n$ times the elementary symmetric functions?