Find the smallest value of a positive integer M such that the polynomial
x⁴- Mx+63 can be written as the product of two nonconstant polynomials with integer coefficients.

(In reply to Solution by Brian Smith)

Your solution is remarkable similar to my approach:  try linear, then quadratic.  But near the end you state:

Therefore for f(x) to have a factorization into two quadratic expressions there must be a factorization of 63 whose two terms sum to a square number.   There are in fact two ways: 1 and 63 or 7 and 9.  In the first case f(x) has a positive M of 64, and in the second case positive M is 16.

For your solution the second case corresponds to f+g = 16 = a^2
Then from the linear terms of your factorization -m = af-ag = -8
so m=8

(and in your first case m=496)

Posted by Jer on 2016-04-03 20:37:25