I'm currently teaching a mini-seminar to high school students, most of whom have at most a background in Algebra/Algebra II (in the US high school system) about finding roots of polynomials.
In general, doing so is rather hard, and by the Abel-Ruffini theorem, there doesn't exist a radical (and field operation) based formula for solving general degree five polynomials and up.
I'd like to convey the intuition for why polynomials are so hard to crack, and here's my idea so far.
When I learned the Abel-Ruffini theorem, my intuition was that for "most" polynomials (with full Galois groups), the roots were so symmetric that they almost "protect" each other: i.e., knowing one will unlock the rest.
As an analogy, a symmetric knot on your earbuds is far harder to untangle than an asymmetric knot as it's hard to know where to start. Or maybe Sudoku is a good analogy, as the unknown cells almost protect each other, and symmetric Sudokus (by which I mean ones in which unknown cells are all tied together) are much harder to solve than ones with "gimme" cells.
As a final analogy, I was thinking about systems of linear equations, and how systems with a few variables as far easier to solve than the rest makes for a much easier system in general.
However, all of the above is handwaving nonsense. Is this intuition meaningful? Is there a way of presenting these ideas without confusing the kids?