# A shift and reduce

**Number Theory**Level 5

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.