nt.number theory - Products of linear forms in 3 variables

This should be a comment to Robin's answer.

Take any irreducible polynomial $f in mathbb{Q}[x]$ of degree 3 with real roots, say $alpha, beta, gamma$. Set $f_1 = x + alpha y + alpha^2 z$, $f_2 = x + beta y + beta^2 z$, $f_3 = x + gamma y + gamma^2 z$.

You can find plenty of polynomials here.