How to use the Division Algorithm (polynomial rings) and evaluation map to prove the Remainder Theorem?

Question

I am trying to use my proof of the Division Algorithm and evaluation map to prove the Remainder Theorem which states:

Let F be a field, a in F, f in F[x]. Then the remainder when f is divided by the polynomial x - a is equal to f(a).

I would like some guidance as to where to start. Thanks in advance.

Answer 1

Write the Euclidean division of $f$ by $(x-a)$
EDIT $$f(x) = Q(x)\cdot\big(x-a\big) + R(x) \textrm{ where } deg(R) < \mathbf{deg(X-a)}$$

What do you get for $f(a)$?