What is a polynomial multiple?

Question

This is probably a really simple thing, but I am unable to find a definition. I tried the internet and various textbooks. This arose from Axler's Linear Algebra done right 3rd. Example 3.103 He says that U is a subspace consisting of all polynomial multiples of x^2. What are all polynomial multiples of x^2? But also in general what is a polynomial multiple? Thank you

Answer 1

An integer multiple of $2$ is any integer times $2$, so any integer of the form $2m$ for some integer $m$. In this case, that just means an even integer.

In the general case, an integer multiple of $n$ is any integer times $n$, so any integer of the form $nm$ for some other integer $m$.

A polynomial multiple of $x^2$ is any polynomial times $x^2$, i.e. $x^2 \cdot q(x)$ for any polynomial $q(x)$. For this example, that just means a polynomial such that its constant coefficient and its degree 1 coefficient both equal zero.

In the general case, given any polynomial $p(x)$, a polynomial multiple of $p(x)$ means any polynomial that can be factored in the form $p(x) q(x)$ for some other polynomial $q(x)$.