Question regarding polynomials and common factors

Question

Let $P(x)$ and $Q(x)$ be distinct polynomials with a common factor $(x-a)$. Show that $R(x)=P(x)-Q(x)$ will have the same common factor.

Answer 1

$P(x)= A(x)\cdot (x-a)$ and $Q(x)=B(x)\cdot (x-a)$ with $A$ and $B$ two polynomials. $R(x) = (x-a)\cdot (A(x) - B(x)) = (x-a) \cdot C(x)$.

Answer 2

By the factor theorem, if x-a is a factor then a is a root (an input that produces 0) . So P(a)=0, Q(a)=0, and R(a)=0-0=0, which by the converse of the factor theorem ( that if and only if a is a root is x-a a factor), then implies R(x) has x-a as a factor.