What is the difference between the remainder theorem and the factor theorem?

The two theorems are similar, but refer to different things. See explanation.

The remainder theorem tells us that for any polynomial ##f(x)##, if you divide it by the binomial ##x-a##, the remainder is equal to the value of ##f(a)##.

The factor theorem tells us that if ##a## is a zero of a polynomial ##f(x)##, then ##(x-a)## is a factor of ##f(x)##, and vice-versa.

For example, let's consider the polynomial

##f(x) = x^2 - 2x + 1##

Using the remainder theorem

We can plug in ##3## into ##f(x)##.

##f(3) = 3^2 - 2(3) + 1## ##f(3) = 9 - 6 + 1## ##f(3) = 4##

Therefore, by the remainder theorem, the remainder when you divide ##x^2 - 2x + 1## by ##x-3## is ##4##.

You can also apply this in reverse. Divide ##x^2 - 2x + 1## by ##x-3##, and the remainder you get is the value of ##f(3)##.

Using the factor theorem

The quadratic polynomial ##f(x) = x^2 - 2x + 1## equals ##0## when ##x=1##. This tells us that ##(x-1)## is a factor of ##x^2 - 2x + 1##.

We can also apply the factor theorem in reverse:

We can factor ##x^2 - 2x + 1## into ##(x-1)^2##, therefore ##1## is a zero of ##f(x)##.

Basically, the remainder theorem links the remainder of division by a binomial with the value of a function at a point, while the factor theorem links the factors of a polynomial to its .