# 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?

Oct 28, 2015

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

#### Explanation:

The remainder theorem says that
the remainder of $f \frac{x}{x - a}$ is equal to $f \left(a\right)$

which implies that $f \left(a\right)$ is a factor of $f \left(x\right)$ only if $f \left(a\right) = 0$

Using the rational factor theorem we know that if $\left(x - a\right)$ is a factor of ${x}^{3} - 5 {x}^{2} - 9 x + 45$
then $a$ is a factor of $45$

We can test all factors of $45$ as indicated below: which tells us $a \in \left\{1 , 3 , - 3\right\}$

So $\left(x - 1\right) , \left(x - 3\right) , \mathmr{and} \left(x + 3\right)$ are all factors of ${x}^{3} - 5 {x}^{2} - 9 x + 45$

Of course multiplicative combinations of these three should also be considered as factors.
For example
$\left(x - 1\right) \cdot \left(x - 3\right) = {x}^{2} - 4 x + 3$ is also a factor.