Extract common factor

Question

Example: g[x_] = (x^3 - 27)/((x - 3)*(x + 1))

Here the common factor is x-3, how can i extract that common factor from a function (g(x)) and put it as a separate ..

Example: f[x] = x-3 

Answer 1

I think PolynomialGCD is the most direct tool:

g[x_] = (x^3 - 27)/((x - 3)*(x + 1));

PolynomialGCD @@ Through @ {Numerator,Denominator} @ g[x]

-3+x

If you don't like operator notation, you might find this version preferable:

With[{n = Numerator[g[x]], d = Denominator[g[x]]}, PolynomialGCD[n,d]]

-3+x