Prove that the subset of polynomial, which is a factor of p(x), is a subspace of P6

Question

Show that: $S=\{p∈P_6:x^2-5x+9 \> \text{is a factor of } p(x)\}$ is a subspace of $P_7$. There is no documentation in my textbook about how to solve this type of question. More specifically I am confused about the "is a factor of p(x)" part. Thanks in advance.

Answer 1

Polynomial $q$ is a factor of polynomial $p$ if there's a polynomial $f$ such that $p=q\cdot f$.

So, $S=\{q\cdot f:f\in P_4\}\,\subseteq P_6$ where $q=x^2-5x+9$ (though it could be any other fixed polynomial).
All you have to prove is that $S$ is closed under linear combinations (or equivalently, under addition and multiplication by scalars).