text: If  is any common divisor of  and , then  divides their GCD.
  for any polynomial . This property is at the basis of the proof of Euclidean algorithm.
  For any invertible element  of the ring of the coefficients, .
  Hence  for any scalars  such that  is invertible.
  If , then .
  If , then .
  For two univariate polynomials  and  over a field, there exist polynomials  and , such that  and  divides every such linear combination of  and  (Bézout's identity).
  The greatest common divisor of three or more polynomials may be defined similarly as for two polynomials. It may be computed recursively from GCD's of two polynomials by the identities:  and
  GCD by hand computation
  There are several ways to find the greatest common divisor of two polynomials. Two of them are:
  Factorization of polynomials, in which one finds the factors of each expression, then selects the set of common factors held by all from within each set of factors. This method may be useful only in simple cases, as factoring is usually more difficult than computing the greatest common divisor.
  The Euclidean algorithm, which can be used to find the GCD of two polynomials in the same manner as for two numbers.