I saw one of my middle school students working on a problem that went something like this:
When $f(x)$ is divided by $x - \frac{1}{2}$, the quotient is $Q(x)$ and the remainder is $R$. What are the quotient and remainder, respectively, when $x f(x)$ is divided by $2x-1$?
Then you had to choose between several answers, each in terms of $Q(x)$ and $R$.
What do you call this kind of problem, and how can I learn how to solve things like this? I never saw anything like this when I did elementary math. For the integer case, it seems like you could use the quotient-remainder theorem, but I don't know how to generalize that to the function case.
Searching for "quotient-remainder theorem with functions" and such has gotten me nowhere.
It's the Polynomial Division Algorithm. The solution involves only scaling that: $\ (a=\frac{1}2$ in OP)
$$\begin{align} \dfrac{f}{x-a} &\,=\, q + \dfrac{r}{x-a}\\[.4em] \Rightarrow\ \ \dfrac{xf}{x-a} &\,=\, xq + \dfrac{xr}{x-a}\\[.4em] &\,=\, xq + \dfrac{(x-a)r + ar}{x-a}\\[.4em] &\,=\, xq+r + \dfrac{\color{#c00}{ar}}{x-a}\\[.4em] \Rightarrow\ \ \dfrac{xf}{2(x-a)}&\, =\, \tfrac{1}2(xq+r) + \dfrac{\color{}{ar}}{2(x-a)} \end{align}\qquad$$
Simpler by congruences: $\bmod x\!-\!a\!:\ \ \begin{align}x\equiv a\\ f\equiv r\end{align}\ \Rightarrow\ xf\equiv \color{#c00}{ar}\ $ by the Congruence Product Rule