Irreducible polynomials: one of them must be of degree zero

Question

I have the following definition of irreducible polynomials, given by my professor:

If $p = f * g$ , where $f, g \in F[x]$ then $degree(f) = 0 \ \ or \ \ degree(g) = 0$

And then the following sentence: it is always possible to represent a polynomial as $p = f*g$, where $deg(f) = 0$ (like this: $p = a * (a^{-1}*p)$)

I am confused, need your help. If every polynomial can be represented as such product with $deg(f) = 0$, given the definition of irreducible polynomials above, doesn't it mean every polynomial is irreducible? (I know, this is false).

F.e., I have a polynomial f = (x+2)(x-5). Well, I can represent it as $a*(a^{-1}*(x+2)(x-5)$ and say that it's irreducible as $deg(a) = 0$.

Do you see what I mean, where my problem is? Would you please help me to understand, what my prof. wanted to tell us?

Answer 1

The confusion was resolved in comments:

• Every polynomial can be factored in a trivial way, i.e., with some of factors being units.
• An irreducible polynomial is one that can be factored only in a trivial way.