The polynomial remainder theorem states that if a polynomial , f(x), is divided by (x -a) using long division of polynomials, the remainder will be f(a). For example if the polynomial is
f(x) = x2 -9 and we divide by x-2, we can work this out by polynomial long division to be
(x2 -9) /(x-2) = x +2 + (-5/(x-2) ) The -5 is called the remainder by analogy to elementary school long division.
We note that f(2) = -5 so the remainder is f(2) as the theorem claims it should be.
Synthetic division could be used instead of long division, but it is easier to think about this in terms of long division if you remember how to do long division of multi-digit numbers.