Question

arrange the following items in the correct order to determine the domain restriction(s) of the rational expression: $f(x)=\frac{3x + 8}{3x^{2}+15x - 18}$ 1. solve for the variable; the domain restriction 2. factor the trinomial 3. set the factors equal to zero 4. set the denominator equal to zero 5. factor the gcf, if needed

Explanation:

Step1: Factor out GCF

First, factor out the greatest - common factor (GCF) from the denominator. For the denominator \(3x^{2}+15x - 18\), the GCF of 3, 15 and 18 is 3. So we have \(3(x^{2}+5x - 6)\).

Step2: Factor the trinomial

Factor the quadratic trinomial \(x^{2}+5x - 6\) inside the parentheses. We get \(3(x + 6)(x - 1)\).

Step3: Set the denominator equal to zero

Since the rational expression \(\frac{3x + 8}{3x^{2}+15x - 18}\) is undefined when the denominator is zero, we set \(3(x + 6)(x - 1)=0\).

Step4: Set the factors equal to zero

We set each factor equal to zero: \(x+6 = 0\) and \(x - 1=0\).

Step5: Solve for the variable

Solving \(x+6 = 0\) gives \(x=-6\) and solving \(x - 1=0\) gives \(x = 1\). These are the domain - restrictions.

Answer:

5, 2, 4, 3, 1