Proof
Let x 
R.
Dividing f(X) by X - x yields
with c of degree at most zero and hence in R.
Evaluating at x yields c = f(x).
Consequently,
f(x) = 0 if and only if X - x divides
f.
Let x R.
Dividing f(X) by X - x yields
with c of degree at most zero and hence in R.
Evaluating at x yields c = f(x).
Consequently,
f(x) = 0 if and only if X - x divides
f.