I was working on some problems and got stumped on this one:
In a polynomial $f(d) = ad^2 + bd + c$. Given there is an integer $d$ for which $f(d)$ has an absolute value greater than $1$. Letting $D = f(d)$, show that $D$ divides $f(Dj+d)$ for every integer $j$.
I understand how this works with an example, but I'm having a hard time figuring out a general proof for this.
This is a part of a bigger question, but I wanted help on this part in particular because I'm not able to proceed without a generalized proof.
Any help would be greatly appreciated!
$$f(Dj+d)=a(Dj+d)^2+b(Dj+d)+c\\=aD^2j^2+2aDjd+\color{green}{ad^2}+bDj+\color{green}{bd}+\color{green}{c}\\=\color{green}{f(d)}+aD^2j^2+2aDjd+bDj$$
Can you show from here that $f(Dj+d)$ is divisible by $D=f(d)$?