This isn't homework or anything, just something I tried and it doesn't seem to be consistent. I randomly just multiplied x^2 - 1 = 0 by x instead of factoring and solving: x^3 - x = 0. Then I factored out an x and got x(x^2 - 1) = 0. Then I divided by x^2 - 1 and got x = 0 which isn't a solution? I know I can just divide by x in x(x+1)(x-1) = 0 but why can't I divide by the other two factors? what am I doing that's algebraically illegal?