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.

Olivier Roche · Answer 1 · 2019-11-12T10:12:48.817

0

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 $f(a)$?

edited Nov 12 '19 at 10:12

answered Nov 12 '19 at 09:34

Olivier Roche

f(a) = q(a) "times" 0 + r(a) = r(a) – asuhdude Nov 12 '19 at 10:01
Oops One had to read "$deg(R)<1$! Hence, $R$ is constant, and you know that $f(a)=R(a)$. – Olivier Roche Nov 12 '19 at 10:13

1 Answers1