SOLUTION: find the domain of R(x)=-2-5x/x^3-5x^2-6x

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Question 346933: find the domain of R(x)=-2-5x/x^3-5x^2-6x
Found 2 solutions by haileytucki, jsmallt9:
Answer by haileytucki(390) (Show Source):
You can put this solution on YOUR website!
(-2-5x)/(x^(3)-5x^(2)-6x)
Factor out the GCF of x from each term in the polynomial.
(-5x-2)/(x(x^(2))+x(-5x)+x(-6))
Factor out the GCF of x from x^(3)-5x^(2)-6x.
(-5x-2)/(x(x^(2)-5x-6))
In this problem 1*-6=-6 and 1-6=-5, so insert 1 as the right hand term of one factor and -6 as the right-hand term of the other factor.
(-5x-2)/(x(x+1)(x-6))
The domain of an expression is all real numbers except for the regions where the expression is undefined. This can occur where the denominator equals 0, a square root is less than 0, or a logarithm is less than or equal to 0. All of these are undefined and therefore are not part of the domain.
x(x+1)(x-6)=0
Solve the equation to find where the original expression is undefined.
x=0,-1,6
The domain of the rational expression is all real numbers except where the expression is undefined.
x≠0,x≠-1,x≠6_(-∞,-1) U (-1,0) U (0,6) U (6,∞)

Answer by jsmallt9(3758) (Show Source):
You can put this solution on YOUR website!
I assume the function is:

Please put numerators and denominators in parentheses in the future. Tutors are less likely to answer your questions if they are not clear.

The domain of R(x) will be all Real numbers except any numbers that make the denominator zero, if any. So we just have to solve:

If a solution exists for this equation it will tell us the numbers that are NOT in the domain. We will solve this 3rd degree equation by factoring it. First factor out the Greatest Common Factor (GCF). The GCF here is x:

Now we factor the trinomial. The factors of -6 that add up to -5 are: -6 and 1. This means the trinomial factors into: (x-6)(x+1). Our equation is now:

From the Zero Product property we know that this (or any) product is zero only if one (or more) of the factors is zero. So:
x = 0 or x-6 = 0 or x+1 = 0
Solving these we get:
x = 0 or x= 6 or x = -1.

So the domain of R(x) is all Real numbers except 0, 6 and -1.