how do i systematically solve
If i use the rational root theorum, then I would factor out either (x+1) or (x+1/2) but when I use those factors to divide into , i get an long confusing answer with remainders.
Use technology.
solve 2x^5 - 5x^2 + 1 = 0 - Wolfram|Alpha
Probably the rational root theorem, which in this case states that the only possible rational roots are , none of which are actually roots. Therefore there are no rational roots.