Finding a polynomial whose roots are connected to the roots of a different polynomial Suppose we have a polynomial function $$f(x) =x^5-4x^4+3x^3-2x^2+5x+1$$ Function $f$ will have 5 roots which can be denoted by $a, b, c, d, e$. I was interested in trying to find a degree 10 polynomial whose roots are given by $abc, abd, abe, acd, ace, ade, bcd, bce, bde, cde$. My idea was that we can relate the coefficients of the degree 10 polynomial to the coefficients of the degree 5 polynomial using Vieta's relations. However, I soon realised that this led to expressions that were extremely difficult to simplify and the method in general, was time-consuming. I was interested in knowing if general techniques exist to solve such problems or if brute is the only way to go about it.
Thanks

$f(x) = x^5-4x^4+3x^3-2x^2+5x+1$
$f$ has $5$ roots donated by $a$, $b$, $c$, $d$ and $e$
The elementary symmetric functions of the roots are
$a+b+c+d+e = 4$
$de+ce+be+ae+cd+bd+ad+bc+ac+ab = 3$
$cde+bde+ade+bce+ace+abe+bcd+acd+abd+abc = 2$
$bcde+acde+abde+abce+abcd = 5$
$abcde = -1$
Let $z = abc$, Computing the elementary symmetric functions of $z$ which are symmetric functions in $a,b,c,d,e$ and expressing them in terms of the elementary symmetric functions of $x$ 
Writing out the conjugates of $z$ shows it's a polynomial of degree $10$
$(z-abc)(z-abd)(z-acd)(z-bcd)(z-abe)(z-ace)(z-bce)(z-ade)(z-bde)(z-cde)$
Expand to express the elementary symmetric functions of $z$
$z^{10}-s_1z^9+s_2z^8-s_3z^7+s_4z^6-s_5z^5+s_6z^4-s_7z^3+s_8z^2-s_9z+s_{10} = 0$
$s_1 = cde+bde+ade+bce+ace+abe+bcd+acd+abd+abc = 2$
$s_2 = {.............}$
This process is large, requires tremendous calculations so I'll skip the details
$s_8 = 
(abcde)^4(cde^2+bde^2+ade^2+bce^2+ace^2+abe^2+cd^2e+bd^2e+ad^2e+c^2de+b^2de+a^2de+bc^2e+ac^2e+b^2ce+a^2ce+ab^2e+a^2be+bcd^2+acd^2+abd^2+bc^2d+ac^2d+b^2cd+a^2cd+ab^2d+a^2bd+abc^2+ab^2c+a^2bc +3( bcde+acde+abde+abce+abcd ) )$
$s_9 = (abcde)^5(de+ce+be+ae+cd+bd+ad+bc+ac+ab) = (-1)^53 = -3$
$s_{10} = (abcde)^6 = 1$
Therefore our polynomial in $z$ is
$z^{10}-2z^9+19z^8-112z^7+82z^6+97z^5-15z^4+58z^3+3z^2+3z+1 = 0$
