Let r be a root of x^3 - 2x + 5 = x^3 - x^2 + 9. Show that none of r, r^2, or r^3 is irrational.
we can actually rearrange this to be a quadratic equation. We move 9,x^2, and x^3 to the left side, and we get that x^2 - 2x - 4 = 0. This is easy to solve with the quadratic formula and we get that the roots are x = 1 + sqrt5, x = 1-sqrt5
we now have that the roots are irrational, the squares of the roots are 6+2sqrt5 and 6-2sqrt5
finally we have that the cubes of the roots are also irrational, 16-8sqrt5 and 16+8sqrt5