But isn't this also true for generic quadratics/cubics too? Like the solution to x^3-2=0 is cubed_root_of(2), so it seems we can "write it down". But what is the definition of cubed_root_of(2)? Well, it's the positive solution to x^3-2=0...
I feel there is an interesting follow-up problem here. The polynomials x^n+a=0 are used to define the "radicals" which is a family of functions F_n such that F_n(a) = real nth root of a = real solution of x^n + a = 0. Using these radicals you can solve all quadratics, cubics and quintics.
Now take another collection of unsolvable polynomials; your example was x^5 - x - 1 = 0 and maybe parameterize that in some way such that these polynomials are unsolvable. This gives us another family of functions G_n. What if we allow the G_n's to be used in our solutions? Can we solve all quintics this way (for example)?
If I'm correct, then the proof would almost certainly use Galois theory!
Nope. Wanna hazard a guess what theory was instrumental in proving that there's no closed form for general quintics?
https://en.wikipedia.org/wiki/Galois_theory#A_non-solvable_q...
One reason to focus on them: when you tell students that x^5 - x - 1 = 0 can't be solved by radicals, that "even if God told you the answer, you would have no way to write it down", this is easy to understand and a powerful motivator for the theory. It's a nice application which is not fundamental, but which definitely shows that the theory has legs.
If you want to know which polynomials are solvable by Newton's method? All of them. It illustrates that Newton's method is extremely useful, but the answer itself is not exactly interesting.