Can I have an explanation of how the remainder function works?

The remainder function is an easy way of determining
whether a term ( x - a) is a factor of a polynomial
P(x).

Whenever P(x) is divided by (x - a), the value of
P(a) is the remainder that P(x) divided by (x - a) would yield. To find out if (x - a)
is a factor of P(x), we substitute x with a. P(a) = 0 indicates that P(x) is divisible
by (x - a)

The proof of the remainder theorem
is:

Take a polynomial P(x); when it is divided by (x – a)
let the quotient be denoted by q and the remainder by r. This gives: P(x) = q*(x – a) +
r. When we substitute x with a, we have P(a) = q*(a – a) + r. Or r =
P(a).

This proves that P(a) is the remainder when P(x) is
divided by (x – a).