Prove that if the reminder of the division W(x) / (x-a) = n, then W(a) = n

Question

I've always taken this property for granted. Only recently have I begun to wonder why this property actually works.
Could you give me some hints how to prove it?

Answer 1

This is really just the definition of "remainder".

The division algorithm tells us, under these conditions, that there is some polynomial $p(x)$ such that $$W(x)=p(x)(x-a)+n$$ If we now take $x=a$ we get $$W(a)=0+n=n$$