1. The problem statement, all variables and given/known data Show that the polynomial f(x)=x^5-3x^4+6x^3+18x^2-3 is NOT solvable by radicals 2. Relevant equations 3. The attempt at a solution I'm pretty sure that to prove that this polynomial is not solvable I am too show that it has exactly 3 roots. That means that it will have 3 roots and 2 complex roots, and thus by a lemma we learned in class the Galois group will be isomorphic to ##S_5## which is not solvable. I am a bit confused on how to prove it has 3 real roots. I guess first I'll take the derivative. f'(x)=5x^4-12x^3+18x^2+36x = x(5x^3-12x^2+18x+36). By the way, this question was a question on last years final, so to solve it I only want to use methods that will be available when I am taking my final in a week (I won't have the general solution to cubic polynomials). So at this point I know that the derivative will have at least 1 real zero, when x=0. Now I'll take a closer look at (5x^3-12x^2+18x+36). This polynomial looks like it may be reducible. I tried to factor it as ##(ax^2+bx+c)(dx+e)## and match coefficients but I could not solve the system of equations, it had 5 variables and 4 equations and I couldn't figure out how to solve it. Anyway, any insight into this from ya'll would be great.