Question 346933
I assume the function is:
{{{R(x) = (-2-5x)/(x^3-5x^2-6x)}}}
Please put numerators and denominators in parentheses in the future. Tutors are less likely to answer your questions if they are not clear.<br>
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:
{{{x^3-5x^2-6x = 0}}}
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:
{{{x(x^2-5x-6) = 0}}}
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:
{{{x(x-6)(x+1) = 0}}}
From the Zero Product property we know that this (or any) product is zero <i>only</i> 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.<br>
So the domain of R(x) is all Real numbers <i>except</i> 0, 6 and -1.