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I was reading Axler's book Linear Algebra Done Right, in which he has a short chapter on vector spaces of polynomials. He is talking about the vector spaces $\mathcal{P}(\mathbb{F})$ which is the vector space of all polynomials with coefficients in field $\mathbb{F}$, where $\mathbb{F}$ is either Real or Complex.

The chapter is really short and even Axler says that the instructor might go over it very quickly, etc.

However, I was looking for a good reference that would dive a little deeper into this subject. The focus of the chapter was on the roots of polynomials, the division algorithm, and the factoring of polynomials such as:

Let $p \in \mathcal{P}(\mathbb{R})$ with degree $m \geq 1$. Then $\lambda$ is a root of $p$ if and only if there is a polynomial $q \in \mathcal{P}(\mathbb{R})$ with degree $m-1$ such that:

$$ p(z) = (z − λ)q(z) $$ for all $z$ in $R$.

I checked Gilbert Strang's book, Freidberg, and a few others, but did not find much coverage at all, if any.

Thanks.

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    Most of the topics you mention aren't really handled by the tools of linear algebra. An introductory text of abstract algebra (especially when covering rings and fields) will probably dive deeper than any linear algebra book would. – 2017-02-22
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    Oh, I see. So I was looking in the wrong place. – 2017-02-22
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    Halmos' Finite Dimensional Vector Spaces book talks about vector spaces of polynomials, though, I haven't worked completely through the text to know the depth. Just some thoughts. I realize this is oldish. – 2017-09-29

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You can find a whole chapter (Chapter 4) on polynomials in the classical text by Hoffman and Kunze, Linear Algebra.

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    Oh great. I will get that one then. I appreciate the suggestion. – 2017-02-22