In the past, I have noticed several problems for which the solution goes something like this:
- Reduce the problem to a polynomial equation
- Find the roots of the polynomial
- Interpret appropriately in the context of the original problem
Some examples that immediately spring to mind are
- Solving linear homogeneous differential equations with constant coefficients
- Finding the eigenvalues of a matrix
- Sketching the frequency response of an LTI system
I have a couple of questions about this:
What's so special about polynomials that makes their roots encode the solutions to such a wide range of problems?
What are some more problems of this type? I would like to compile a big list of these.
Answering the question of why polynomials are such a universal object is not easy. It is remarkable how often they show up in all areas of math.
My favorite example of polynomials doing cool things is the Cyclotomic Polynomials whose roots are the primitive roots of unity. I really like this example because Mobius inversion gives us an explicit formula for the cyclotomic polynomials.
At the bottom of this Wikipedia article is a list of named polynomials that come up in various areas of math.