What is the difference between the remainder theorem and the factor theorem?
The two theorems are similar, but refer to different things.
The remainder theorem tells us that for any polynomial
The factor theorem tells us that if
For example, let's consider the polynomial
Using the remainder theorem
We can plug in
Therefore, by the remainder theorem, the remainder when you divide
You can also apply this in reverse. Divide
Using the factor theorem
The quadratic polynomial
This tells us that
We can also apply the factor theorem in reverse:
We can factor
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 zeros.