GCD of f(x) and f'(x) (simple question)

This seemed simple, but I keep getting ridiculous fractions.

For example, find the GCD of f(x) and f'(x) over Q of:

f(x)=x^3 -3x -2

The quotient becomes 1/3x with a remainder of -2x-2, which can be simplified to x+1.

From my understanding, x+1 is the GCD. This doesnt seem correct, taking the quotient into consideration. Can anyone check this?

As a sample, my book has GCD(f(x),f'(x))= x-1 when f(x)=x^4-x^3-x+1. I could not work this question out!

Also, as a very similar question, if I get a remainder of -2x^2+x, can I simplify this to -2x+1, or does such a simplification not work for GCD's?