A shift and reduce
For how many positive integers \(N<1000\), does there exist a polynomial \(f(x)\) with integer coefficients such that all of the following conditions are true:
The degree of \(f\) is at most \(6.\)
The sum of absolute values of coefficients of \(f\) is at most \(7.\)
The polynomial \(f(x)+N\) is reducible over the integers.
Details and assumptions
A polynomial that is reducible over the integers can be expressed as a product of two non-constant polynomials with integer coefficients.