Problem:
Find the smallest positive integer n with the property that the polynomial x4−nx+63 can be written as a product of two nonconstant polynomials with integer coefficients.
Solution:
If ax−b is a factor of the given polynomial, then a=1 and b is a root. Thus n=b3+b63​, which achieves a minimum integer value of 48 when b=3. On the other hand, suppose
x4−nx+63=(ax2+bx+c)(dx2+ex+f)
where all coefficients are integers. By multiplying both factors by −1 if necessary, it can be assumed that a>0; thus ad=1 implies a=d=1. Equating coefficients for x3 implies that b+e=0, so
x4−nx+63=x4−(c+f−b2)x2−(bc−bf)x+cf=x4+(c+f−b2)x2+b(f−c)x+cf.
The coefficient of x2 is c+f−b2, and the constant term is cf=63. Thus c+f=b2, and so c and f are positive. The pairs of positive factors of 63 sum to 1+63=64,3+21=24,7+9=16,of which only the first and the last are squares. In the first case, b=±8, and
(x2±8x+63)(x2∓8x+1)=x4∓496x+63
In the second case, b=±4, and
(x2±4x+9)(x2∓4x+7)=x4∓8x+63.
Thus the smallest possible value of n in this case is 8​, which is less than the value in the previous case and hence the minimum.
The problems on this page are the property of the MAA's American Mathematics Competitions