Beware - one step more and you get into the region of generating functions. I recommend a book Herbert Wilf with a wonderful name of Generatingfunctionology (https://www2.math.upenn.edu/~wilf/gfology2.pdf).
You can also multiply polynomials by way of analogy with integer multiplication:
3 1 2 1
× 2 0 6
------------
18 6 12 6
0 0 0 0
6 2 4 2
-----------------
6 2 22 8 12 6
The visualizations make the concept easy to grasp.
This and complex analysis are fascinating topics in Undergraduate studies.
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My favourite use case for this: By the same derivation as this blog, one can prove that, if you have any two probability distributions X and Y (they can be different), the probability distribution of X+Y is a convolution of the PMFs/PDFs of X and Y.