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

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Answer 1 · 2015-10-28T21:00:34

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

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:
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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.

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+45x^3-5x^2-9x+45?