Artin Textbook Proposition 12.3.7(a):
Let f be an integer polynomial with positive leading coefficient. Then f is an irreducible element of Z[x] if and only if it is either a prime integer or a primitive polynomial that is irreducible in Q[x].
The proof is divided by cases.
For me, I only do not understand partially.
The question I have is for the case f is non-constant: from the proof given, f has an integer factor different from +1 or -1, since f is irreducible over Z[x], which makes sense. But I do not know "so if its leading coefficient is positive, it will be primitive".
I asked my instructor, the feedback is "For a non-constant polynomial, an integer factor need not be +1 or -1. For example, 2x+4 has a factor of 2."
I do not understand what does it means. Anyone could help? Thanks advance.
Note: Regarding the book says: "If f is irreducible and not constant, it cannot have an integer factor different from +1 or -1, so if its leading coefficient is positive, it will be primitive".