How am I sure that factoring a root from a polynomial in $R[x]$ remains in $R[x]$? How am I sure that factoring a root from a polynomial in $R[x]$ remains in $R[x]$?
I am sure I studied this during my Ph.D., but the argument escapes me.
SO, suppose we are looking at $f(x) \in R[x],$ with $r \in R$, and $f(r)=0$. How do we know that $g(x)=f(x)/(x-r)$ is in $R[x]$? What is the general result around this area?

Let $f=a_0+a_1X+\dots+a_nX^n$ with $f(r)=0$. Now consider the polynomial
$$\begin{align}\tilde f&=f(X+r)\\&=a_0+a_1(X+r)+\dots+a_n(X+r)^n\\&=b_0+b_1X+\dots+b_nX^n.\end{align}$$
The coefficients $b_i$ can be obtained by multiplying out all the parentheses. This is a polynomial with $\tilde f(0)=0$. But $\tilde f(0)=b_0$, so $b_0=0$ and
$$\begin{align}\tilde f&=b_1X+\dots+b_nX^n\\
&=X(b_1+\dots+b_nX^{n-1}).\end{align}$$
Now since $f=\tilde f(X-r)$, we now have
$$f=(X-r)[b_1+\dots+b_n(X-r)^{n-1}],$$
where the polynomial in straight brackets is $g$. Note that all the polynomials and factors involved have coefficients in $R$.
