The remainder theorem stated that " If a polynominal f(x) is divided by (x-k), the remainder is f(k)."
So I think of a constant function ,say, f(x)= 5 and I divide this by (x-1). According to this theorem, the remainder must be f(1). So, f(1)= 5 which just returns the original input value as it is a constant function.
But in reality, let's say for the situation f(5)= 5, if I divide 5 by 4, which also is f(x) divided by (x-1), I get the remainder 1, that contradicts the theorem. So I just want to know whether this theorem really doesn't work for constant function and if there are also other situations where this theorem won't work. Thanks
Edit: I have read that the remainder is always constant and does not depend on the value of "x" in f(x) but I'm curious why this happens.