The remainder theorem says when a polynomial function is divided by ax-b, the remainder is f(b/a). But if you take a function and divided it by 2x-1, for example, the remainder will be obviously different than when divided by 4x-2, which is 2x-1 multiplied by two. f(2/4) is the same as f(1/2), so the result when subbed into f(x) should be the same, but their remainders are obviously different. Can anybody clarify?

Thanks in advance!