Classroom
When you take a linear algebra course and get to the chapter on eigenvalues, your homework problems will include a small matrix A and you will be asked to find the eigenvalues. You do this by computing the determinant
det(A − λI) = P(λ)
and getting P(λ), a polynomial in λ. The roots of P are the eigenvalues of A.
Either A will be a 2 × 2 matrix, in which case you can find the roots using the quadratic formula, or the matrix will have been carefully selected so that P(λ) will be easy to factor. Otherwise, finding the roots of a polynomial is hard.
Real world
Numerical algorithms to find eigenvalues have gotten really good. In practice, you don’t compute determinants or find roots of polynomials. Instead you do something like the QR algorithm.
Finding all the roots of a polynomial is a challenging problem, and so what you might do in practice is find the roots by constructing a matrix, called the companion matrix, whose eigenvalues correspond to the roots you’re after.
Summary
As a classroom exercise, you calculate roots of polynomials to find eigenvalues.
In the real world, you might use an eigenvalue solver to find the roots of polynomials.
I wrote a similar post a few years ago. It explains that textbooks definite hyperbolic functions using ex, but you might want to compute ex using hyperbolic functions.