Theorem

Every non-constant polynomial $f\in\mathbb{F}[x]$ can be written as a product of a unit and irreducible polynomials. This factorization is unique up to associates. i.e. if $$f=\mathscr{l}_{1}\dots \mathscr{l}_{k}=\mathscr{l}_{1}'\dots \mathscr{l}_{t}'$$ are two different factorizations of $f$ as a product of irreducible polynomials $\mathscr{l}_{i}$ and $\mathscr{l}_{j}'$ with $1\le i\le k$ and $1\le j \le t$, then $k=t$ and (after a permutation) the irreducible factors $\mathscr{l}_{i}$ and $\mathscr{l}_{i}'$ are associates.

Corollary

We have that for any $f\in\mathbb{F}[x]$, $f$ can be factored as $$f(x)=u\prod_{i=1}^{k}\mathscr{l}_{i}(x)$$where $u$ is a unit and $\mathscr{l}_{i}$ irreducible. If $f$ is monic then $$f(x)=\prod_{i=1}^{k}\mathscr{l}_{i}(x)$$with each $\mathscr{l}_{i}$ a monic irreducible polynomial.

Remark

Note that $$\deg f=\sum_{i=1}^{k}\deg \mathscr{l}_{i}$$