HCF of polynomials

Question

If hcf(P,Q)=hcf(P,R)=1 then hcf(P,QR)=1 ,how would I try a prove this, I tried using Bezout's Lemma to get APR-R=-B(QR), but how would I proceed?

Answer 1

One possibility, using Bézout, is to find $s, t, u, v$ such that $$ s P + t Q = 1 = u P + v R, $$ and then compute $$ 1 = s P + t Q = s P + t Q \cdot 1 = s P + t Q (u P + v R) = \dots $$