If $f$ is not primitive, then it is not irreducible?

Question

This is the beginning of a proof from Algebra, Chapter $0$ by Aluffi.

I am not sure why this statement is true. If $f$ is not primitive, then the gcd of its coefficients is not $1$. But what if the content of $f$ is a unit. Then we can factor $f$ but it would still be irreducible. What am I missing?

Answer 1

This is mainly echoing the comment above, but I think what you're missing is the definition of primitive. The definition of primitive is not that the content is $1$, it's that the content is a unit.