Let a, b, c, d be distinct integers such that
(xa)(xb)(xc)(xd)  4 = 0 has an integer root r.
Show that 4r=a+b+c+d.
The expansion of the function starts with the two terms:
x^4  (a+b+c+d)x^3 + ....
So the sum of the four roots is a+b+c+d.
The mean of the four roots is (a+b+c+d)/4.
Roots tend to occur in pairs of the form mean_of_roots ± something.
I am guessing that for this function the "something" will be complicated enough such that for the root to be an integer, the something would have to be zero making r = (a+b+c+d)/4.

Posted by Larry
on 20240901 13:04:15 