I'm given a polynomial of degree $4$ and its roots, let's call them $r_1,$ $r_2$, $r_3$ and $r_4$. I'm asked to show what is the value of the expression of $\sum r_1^2r_2$, that is, the sum of all different monomials one can form with permutations of the variables on the expression $r_1^2r_2$.
I know the relation between a polynomial an its roots. So, I know what is the value of the elementary symmetric polynomials $s_i$ and I just need to express all the polynomial $\sum r_1^2r_2$ in terms of $s_i$. There is a generic procedure to do so which I take from Cox's book "Galois theory".
However, I see that computing the value by hand apparently takes a lot of time. So for the first summand I should take the symmetric polynomial $f-s_1s_2$. Then I should repeat the process until I get to a zero polynomial.
I think there is a way to speed up the computations based in a form of symbolic computation. For instance in page 33, Cox reasons in the following way:
Given $f = \sum_4 x_1^3x_2^2x_3$ the leading term of $f$ is $x_1^3x_2^2x_3$ and therefore we should use polynomial $s_1s_2s_3$. Then $f_1 = f - s_1s_2s_3$ and here comes the magic:
$\sum s_1s_2s_3 = \sum_4 x_1^3x_2^2x_3+3\sum_4x_1^3x_2x_3x_4+3\sum x_1^2x_2^2x_3^2+ 8 \sum_4x_1^2x_2^2x_3x_4$
It is true that Cox claims that maybe a computer is helpful for this case. But how could I calculate this expression in order to avoid to directly calculate all the monomials and doing all the calculations?