When, where and **how often** do you find polynomials of higher degrees than two in mathematical, pure/applied, research?

The characteristic polynomial of an $n \times n$ matrix has degree $n$. We often care about matrices larger than $2 \times 2$.

It's actually (somewhat) rare in research that you are lucky enough to have a polynomial that is only degree one or two. Some examples I haven't seen mentioned already: cyclotomic polynomials, irreducible polynomials used to construct field extensions, all sorts of polynomials used to construct algebraic curves used in cryptography.

While we don't always have nice formulas to explicitly find the roots, we have other ways to work with them. For example, we can choose nice polynomials that have a special form, or construct a polynomial so we already know the roots. We can use computational techniques to approximate the roots. We can use them in applications where we don't care what the roots are.

Cubic polynomials are ubiquitous in computer-aided design and computer graphics.

They also are the basis for computer fonts.

Finite element analysis is based on polynomial functions. Isogeometric analysis uses NURBS.