For p,q,r,s,a,b,c,d in Z-{0}, suppose px³ + qx² + rx + s = 0.
If px³ + qx² + rx + s can transform (ax - b)(cx - d)(ex - f),
we get
px³ + qx² + rx + s = (ax - b)(cx - d)(ex - f)
= acex³ + … -bdf
Compared with both sides, p = ace, s = -bdf.
Therefore,
s/p = -bdf / ace,
while the solutions of (ax - b)(cx - d)(ex - f) = 0 are
x = b/a, d/c, f/e.
Thus the candidates of solutions of the equation lies in s/p = -bdf / ace.