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#?

1 Answer
Alan P.
Oct 28, 2015

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:
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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.

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