The theorem stating that if a polynomial in x, f(x), is divided by (x - a), where a is any real or complex number, then the remainder is f(a).
For example, if f(x) = x2 + x - 2 is divided by (x-2), the remainder is f(2) = 22 + (2) - 2 = 4. This result may become obvious if we change the polynomial into an equivalent following form:
f(x) = (x-2)(x+3) + 4
As shown, the above expression can easily lead us to expect 4 as the remainder when f(x) is divided by (x-2).
The remainder theorem can help find the factors of a polynomial. In this example, f(1) = 12 + (1) - 2 = 0. Therefore, it means that there is no remainder, that is, (x-1) is a factor. This can be easily shown once we rearrange the original polynomial into the following equivalent expression:
f(x) = (x-1)(x+2)
As shown, (x-1) is a factor.