A statement that is equivalent: if the hcf for P and for Q is 1, then it is 1 for R, also.
In the case of one variable there is a simple proof of this. Consider a prime number p, and try to show that R mod p (i.e. R with coefficients reduced to the field of residues modulo p) is not 0. In fact the degree of R mod p is the sum of those of P mod p and of Q mod p, which is more than enough, because we are working in a field.
An important consequence is that R can only factorise as a product of polynomials with rational number coefficients, if it already does into integer polynomials. One sees this by checking the powers of a fixed prime p needed to clear denominators; the same argument works as before, and this version can also be called the Gauss lemma. It applies to the rational root theorem.
There is a generalisation to several variables.