SOLUTION: I am trying to figure out if the denominator of this fraction factors any lower than what it is. The fraction is 3x^3+x^2-27x-9/ 2x^3+9x^2-8x-15. I know that I can't factor by grou

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Question 378418: I am trying to figure out if the denominator of this fraction factors any lower than what it is. The fraction is 3x^3+x^2-27x-9/ 2x^3+9x^2-8x-15. I know that I can't factor by grouping because the groups in this polynomial in the denominator wouldn't have anything in common. I am stuck on this problem and don't know what else to try, please help!!
Answer by solver91311(24713) (Show Source):
You can put this solution on YOUR website!

First use the Rational Roots Theorem which says that if

has a rational root then it must be of the form:

Where is an integer factor of and is an integer factor of

Using that, the only possible rational factors of

are

Use synthetic division to test these 16 trial divisors until you either find one that divides the original cubic polynomial evenly, leaving you with the successful trial divisor as one factor and a quotient consisting of a quadratic trinomial as another (and hopefully further factorable) factor, or you exhaust all of the possibilities and thereby determine that your denominator is prime.

If you need a refresher on Synthetic Division, see:

Purple Math: Synthetic Division (There are 4 pages -- read them all).

Write back and let me know how you make out.

John

My calculator said it, I believe it, that settles it