(Solved): 3. Suppose \( R \) is a UFD (unique factorization domain) (a) Define what it means for a polynomial ...

3. Suppose \( R \) is a UFD (unique factorization domain) (a) Define what it means for a polynomial \( f(x) \in R[x] \) to be a primitive polynomial (b) Supppose \( f(x) \in R[x] \) with \( \operatorname{deg} f(x) \geq 1 \) such that \( f(x) \) is irreducible. Prove \( f(x) \) must be primitive (you can assume gcds exist in UFDs) (c) In the ring \( \mathbb{Z}[x] \) prove that the product of primitive polynomials is primitive as follows: assume \( f(x), g(x) \) are primitive but \( f(x) g(x) \) is not primitive. Then, there is a prime \( p \) in \( \mathbb{Z} \) such that \( p \mid f(x) g(x) \). Now, use the reduction of coefficients \( \bmod p \) homomorphism \( \pi_{p}: \mathbb{Z}[x] \rightarrow \)