Let a, b, c, d be distinct integers such that
(x-a)(x-b)(x-c)(x-d) - 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.
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Posted by Larry
on 2024-09-01 13:04:15 |
a first step
