Verified answer

if X - r divides f(X) then f(r) = 0

indeed

f(X) = (X - r)Q(X) (remainder is 0)

f(r) = (r - r)Q(r) = 0

a0 + a1r ... + r^n = 0

a1r ... + r^n = - a0 (*)

if r is not integer and it is rational, also all of its powers were not integer,

even if all a_i (i=1,..n-1) simplifies with r^i giving integer result in (*)

r^n is not integer and cannot be equal to - a0 which is integer

his is absurd so r must be integer