Question

How do you use the remainder theorem to find which if the following is not a factor of the polynomial `x^3-5x^2-9x+45`?

Answer 1

Since you did not supply the list referenced as "the following"
I will supply the binomial factors:
`(x-2), (x-3), (x+3)`

Explanation:

The remainder theorem says that
the remainder of `f(x)/(x-a)` is equal to `f(a)`

which implies that `f(a)` is a factor of `f(x)` only if `f(a)=0`

Using the rational factor theorem we know that if `(x-a)` is a factor of `x^3-5x^2-9x+45`
then `a` is a factor of `45`

We can test all factors of `45` as indicated below:

which tells us `a in {1,3,-3}`

So `(x-1), (x-3), and (x+3)` are all factors of `x^3-5x^2-9x+45`

Of course multiplicative combinations of these three should also be considered as factors.
For example
`(x-1)*(x-3) = x^2-4x+3` is also a factor.