Proving a property of two polynomials when one of them divides the another

Question

Suppose $P(x)$ is a polynomial which can be factored into a product of different linear terms, that is $P(x)=(x-\alpha_1)(x-\alpha_2)\cdots(x-\alpha_k)$ and suppose $Q(x)$ divides $P(x)$ (i.e. $Q \mid P$). How one can prove that $Q(x)$ can also be factored into a product of different linear terms (and even stronger: such that all of them are the terms which appear in $P(x)$ factorization)? This statement appeared in my linear algebra course but I fail to prove it for myself.

Answer 1

Hint.

An irreducible element that divides $Q$ divides $P$.

Answer 2

Hint $\ $ Assuming the coefficient ring is a domain, then each factor $\,x-a_i\,$ is prime, and prime products always factor uniquely; in particular the factors of $\,p_1^{e_1}\cdots p_k^{e_k}\,$ are of form $\,p_1^{f_1}\cdots p_k^{f_k}\,$ for $\,f_i \le e_i\,$ (up to associateness, i.e. unit factors).