Understanding the quotient ring $\mathbb{R}[x]/(x^3)$.

Let $p = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \cdots + a_n x^n \in \mathbb{R}[x]$. Then the image of $p$ under the natural map $\mathbb{R}[x] \to \mathbb{R}[x]/\langle x^3 \rangle$ is $a_0 + a_1 x + a_2 x^2 + \langle x^3 \rangle$.

I.e., every element in $\mathbb{R}[x]/\langle x^3 \rangle$ is of the form $$(\text{polynomial of degree $2$}) + \langle x^3 \rangle$$

Elements of $\mathbb{R}[x] / (x^3)$ are real polynomials... modulo $x^3$.

Sure, that's nearly an empty statement, but it is a very general and useful view on quotient objects of any sort -- the elements of the quotient are "named" by elements of the original object, and the relation (here, congruence modulo $x^3$) tells us when two names are equal.

However, for many quotient rings, you can choose canonical (or otherwise reduced in some sense) representatives. A familiar example is that each element of $\mathbb{Z} / (47)$ has a canonical representative as an integer in the interval $[0, 46]$. And it can be computed efficiently through division with remainder.

Univariate polynomial rings (over fields) are similar -- division with remainder gives us a way to select canonical representatives in its quotient rings. Since $x^3$ has degree three, every element of $\mathbb{R}[x] / (x^3)$ has a canonical representative of degree $\leq 2$.

In this specific case, in the way polynomials are usually represented, the representative is especially easy to compute: just truncate!

$$a + bx + cx^2 + dx^3 + \ldots \equiv a + bx + cx^2 \pmod{x^3}$$

For multivariate polynomial rings, you need the idea of a Gröbner basis to do similar things.

As a general principle, univariate polynomial rings over fields (especially finite fields) are extremely similar to the integers in terms of their algebraic properties; most notions you have about one sort can be translated to the other.

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Possibly interestingly, another interpretation is available here. We can view elements of $\mathbb{R}[x] / (x^3)$ as second-order approximations to real power series (whether you mean formal power series, or the notion of convergent power series from calculus).

So, we can view its elements as analytic functions, where we allow ourselves to have errors in the third order.

This is sometimes rather useful. I've seen some calculations greatly simplified in $\mathbb{R}[x] / (x^2)$ by rewriting

$$ 1 + a x \equiv \exp(ax) \pmod{x^2} $$

and other similar sorts of things, although I don't recall any examples at the moment.

You would imagine $\mathbb{R}[x]/(x^3)$ as a ring formed when you add to $\mathbb{R}$ an abstract element, $y$, that satisfies the relation $y^3=0$.