Artin defines the adjoining of an element $\alpha$ to ring $R$ satisfying polynomial $f \in R[x]$ by $$R[\alpha] = R[x]/(f).$$
If $f$ is monic with degree $n$, then we get some nice properties, such as a unique representation of any element of $R[x]/(f)$ as a $n-1$-degree polynomial in $\alpha$. $$\sum_{k=0}^{n-1} c_k \alpha^k$$ This follows from division with remainder by a monic polynomial.
However, what happens when $f$ is not monic? I am having trouble figuring out what still holds. Here's my guess: the elements of $R[x]/(f)$ are still polynomials in $\alpha$, but the representation is no longer unique, and we may need polynomials of arbitrarily large degree (in contrast with the $n-1$-degree polynomial above).