Divisibility of polynomials is a much more rigid property than in integers. Given an integer $n$, another $N$ has a non-zero probability $\frac1n$ to be a multiple of $n$. On the contrary, the probability (no rigour here) that a ``random'' polynomial $P\in{\mathbb C}[X]$ be a multiple of a given one $q$ is zero. For instance, $P$ is a multiple of $X$ iff $P(0)=0$, an event of probability $0$.
If you increase the structure, by either considering multi-variate polynomials, or polynomials with coefficients in $\mathbb Z$, you ``simplify'' even more, in the sense that you have additional criteria for divisibility of primality (Newton's polygon, Eisenstein's criterion, ...), and you have a huge theory (Galois' theory) which you can use and abuse.
Edit. An example of the powerness of polynomials that I like a lot is the following.
Theorem. Let $k$ be a field and $A_0,A_1,B_0,B_1\in M_n(k)$ be given, with $A_0,A_1$ invertible. Let $X$ be an indeterminate. If $XA_0+B_0$ and $XA_1+B_1$ are equivalent in $M_n(k[X])$, the there exist $G,H\in GL_n(k)$ such that $GA_0=A_1H$ and $GB_0=B_1H$.
The case where $A_0=A_1=I_n$ is at the basis of the theory of similarity invariants.