Suppose I had the quintic equation $(x+1)(x+2)(x+3)(x+4)(x+5)=0$. Does the insolvability theory mean that I can only get approximations because the root is an generally an irrational number, or does it mean that even in this case I can only get an approximation to say -5 as well even though it is whole number and is an exact root?
I've read that Galois theory can tell you if a quintic polynomial is the type that can be solved exactly. My question is: Are these types of quintics simply the ones with integer roots? I would suspect that just because a quintic has integer coefficients doesn't mean they have integer roots.
Finally, If galois theory says that a particular quintic is of the exactly solvable type, what method is used to solve them exactly, besides the rational root theorem? I'm pretty sure it would be something like Cardano's method, adapted to a fifth degree equation. Where can you find this method?