Rest theorem and rest function

[pseudo beacon]

I was doing some problems with a method in my textbook. It however, seems to assume that the rest funciton, $r(x)$ in $p(x) = k(x)q(x)+r(x)$for some polynomial $p(x)$ is the same, no matter what polynomial $k(x)$ you divide $p(x)$ with. Why is this so?

[limpid palm]

Coleslaw

[candid salmon]

Can you show what you mean by this? Maybe you only think it assumes it’s the same