Theorem: An element a ∈ F is a root of polynomial P ∈ F[X] if and only if (X-a) divides P.
(1) deg(X - a) =1 [See Definition 4, here for definition of degree]
(2) Therefore, the remainder R of the division of P by (X - a) is a constant polynomial. [See Theorem, here]
(3) So, from the Division Algorithm for Polynomials (see Theorem, here), there exists Q,R such that:
P = (X - a)Q + R
P(a) = (a -a)Q + R = R
(5) This shows that P(a) = 0 if and only if R = 0. That is, P(a) = 0 if and and only if P is divisible by (X - a).
- Jean-Pierre Tignol, Galois' Theory of Algebraic Equations, World Scientific, 2001