What does it mean to multiply or divide polynomials?
I am a bit uncertain by what you mean by graphical. One notion is to plot or graph a function, like $y=f(x)$, as you would in a highschool level algebra course. Another notion of "graphical" may include some similar to a Cayley graph. I just want to be upfront that the extent of my knowledge does not extend greatly into topology and some algebraic geometry. I will go off on the assumption that overly abstract "graphical" representations do not count.

Graphical Plots: In abstract algebra, the polynomials that work with the $y$ vs. $x$ type of graphing are in the ring of polynomials over $\mathbb{R}$, otherwise known as $\mathbb{R}[x]$. Evaluating any of these for an $x \in \mathbb{R}$ gives you a continuous function that you can plot. This works for $\mathbb{R}[X]$'s extensions, too ($\mathbb{C}[X]$).

Graphs: Cryptography and coding theory involve a massive number of structures, each of which can be represented as a graph. A notable set of examples are graphs depicting algebras over structures of finite size. Rings of polynomials are not one of them, since they contain, at minimum, a countable number of elements. Some intuition can be gathered from how these graphs change as you vary/limit the degree of your polynomial (more on this later). Though I imagine that this is not the kind of representation that you had hoped.

For other graphical/geometrical/topological aspects of crypto and coding theory, you might want to look into things like vector spaces, sphere packing, and lattices.

Polynomials Over Finite Fields:

Even so, it is important to realize that the inclusion of rings of polynomials in cryptography and coding theory are over finite fields. That is, they usually only consider $\mathbb{F}_q[X]$, which is the finite field with $q$ elements. An easily recognized one would be $\mathbb{F}_2^k = \{0,1\}^k$ (your typical algebra over binary strings).

A slight extension of this idea, rings of multivariate polynomials, are also useful. They are typically represented by $\mathbb{F}[X_1,X_2,...]$ where $\mathbb{F}$ is a field. An algebra of multivariate polynomial over $\mathbb{F}_2$ is equivalent to a boolean algebra. A boolean formula, $\varphi(x_1,x_2,...,x_n)$, would be equivalent to evaluating a degree $n$ multivariate polynomial over $\mathbb{F}_2$. The intuition behind this is to see that $+$ and $\times$ act like XOR and AND logic gates on your binary inputs, $x_1,x_2,...,x_n \in \{0,1\}$.

Beyond drawing something fundamentally equivalent to circuit diagrams, I do not see a fruitful way to represent this graphically. Though, this did not stop these guys from trying (I still think it is the same): http://polybori.sourceforge.net/zdd.html

There are obviously way more concepts from abstract algebra that are involved in these areas (e.g. $\mathbb{Z}_p$, the ring integers mod $p$). Some may even have graphical representations that make the idea more intuitive. Though these fall outside the scope of the question, which is specific to algebras over polynomials.