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

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 zeros.