In Charles Pinter's Abstract Algebra, he presents the factorization theorem for polynomials. That is,
Every polynomial $a(x)$ of positive degree in $F[x]$ can be written as a product \begin{equation*} a( x) =kp_{1}( x) p_{2}( x) \cdots p_{r}( x) \end{equation*} where $k$ is a constant and $p_1(x),...,p_r(x)$ are monic irreducible polynomials of $F[x]$.
But I am confused by the next paragraph,
If this were not true, we could choose a polynomial $a(x)$ of lowest degree among those which cannot be factored into irreducibles. Then $a(x)$ is reducible, so $a(x)=b(x)c(x)$ where $b(x)$ and $c(x)$ have lower degree than $a(x)$. But this means that $b(x)$ and $c(x)$ can be factored into irreducibles, and therefore $a(x)$ can also.
My issue is if we choose a polynomial $a(x)$ among those which cannot be factored into irreducibles then how is $a(x)$ reducible? We just said choose one that isn't reducible. Moreover, I don't understand how we immediately know that $b(x)$ and $c(x)$ are reducible as well. All we said was that they were of lower degree, not that they were reducible.
This leads to the more general question, how can every polynomial of positive degree be factored into irreducibles? For example, take any one of these irreducible factors $p_r(x)$, by definition, $p_r(x)$ cannot be written as a product of irreducibles since this would mean that it is reducible and thus couldn't be an irreducible factor of $a(x)$. It seems to me that we are saying that even irreducible polynomials can be written as a product of irreducibles. I can't seem to locate where my misunderstanding is.
The point you need to understand is that factoring a polynomial into a product of a single term is allowed. Indeed, this is the only option for irreducible monic polynomials.
In this way, if $a(x)$ were irreducible, then writing $a(x)$ would be a product of irreducible factors (which is assumed not to exist), so $a(x)$ must be reducible. As for $b(x)$ and $c(x)$, they are not necessarily reducible, and could just be a product each of one term.