Simply in what sense? Removing common factors?
There are only 8 things to try to find a rational root of the numerator. It shouldn't take 30 minutes to work out what those 8 are and to test them. (note that it can be proven that if they have a common factor, then not only does the factor have integer coefficients, but both cubics also have a rational root)
Or, you can use the Euclidean algorithm to find the (polynomial) greatest common divisor of your two polynomials.
e.g. to find the gcd of $x^2 - 2x + 1$ and $x^2 - 1$, you can compute:
$ \begin{align} (x^2-2x+1, x^2-1) &= (x^2 - 2x + 1 - (x^2 - 1), x^2 - 1) \\ &= (-2x + 2, x^2 - 1) \\ &= (x-1, x^2 - 1) \\ &= (x-1, x^2 - 1 - x(x-1)) \\ &= (x-1, x-1) \\ &= x-1 \end{align}$
(note the "size" of a polynomial here is determined by its degree, and I was "lazy" and avoided doing a full division-with-remainder calculation)