1. The problem statement, all variables and given/known data Show that the polynomial f(x) = x^5 - x^3 - 3x^2 + 3 is solvable by radicals where the coefficients of f are from the field of rational numbers. 2. Relevant equations 3. The attempt at a solution My strategy to solve this problem was to construct a splitting field and then see if that splitting field lies in some radical extension. I first noted that 1 is a root of f(x), so I divided f(x) by (x-1) and got the quotient x^4 + x^3 - 3x - 3. From here I hit a block. I tried assuming r to be a root and then divided x^4 + x^3 - 3x - 3 by (x-r) to arrive at the quotient x^3 + (r+1)x^2 + (r^2+r)x + (r^3 + r^2 - 3) which didn't seem to be much help. How else can I try to find the roots of x^4 + x^3 - 3x - 3 so I can construct a splitting field?