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