Finding the roots of a linear polynomial is trivial. Already the Babylonians could find roots of quadratic polynomials. Methods to solve cubic polynomials and fourth-degree polynomials were discovered in the sixteenth century, all using radicals (i.e. $n$th roots for some $n$). Isn't it obvious that finding the roots of higher degree polynomials is also possible using radicals and that we have not found the formulas yet is only because they become more and more complicated with higher polynomial degrees?
Galois theory shattered this belief.