Let p(x) be the polynomial (x - a)(x - b)(x - c)(x - d). Assume p(x) = 0 has four distinct integral roots and that p(x) = 4 has an integral root k. Show that k is the mean of a, b, c, d.
Solution
a, b, c, d must be integers. (k - a)(k - b)(k - c)(k - d) = 4 and all of (k - a), (k - b), (k - c), (k - d) are integers. They all divide 4, so they must belong to {-4, -2, -1, 1, 2, 4}. They are all distinct, so at most two of them have absolute value 1. Hence none of them can have absolute value 4 - or their product would be at least 8. Hence they must be -2, -1, 1, 2. Hence their sum is 0, so 4k = a + b + c + d.
© John Scholes
jscholes@kalva.demon.co.uk
5 Mar 2002