Problem 104 - College Algebra Essentials

Question

List all numbers that must be excluded from the domain of each expression. $$\frac{x^{3}-2 x^{2}-9 x+18}{x^{3}+3 x^{2}-x-3}$$

Step-by-Step Solution

Answer

The numbers that must be excluded from the domain of the expression are \(x=1\), \(x=-1\), and \(x=-3\).

Step 1: Set the denominator equal to zero

Find the values of x for which the denominator becomes zero by setting it equal to zero: \(x^{3}+3x^{2}-x-3=0\)

Step 2: Factor the cubic equation

Factor the equation to solve for the roots: This can be done using the rational root theorem or synthetic division, or by using a calculator if it's allowed. The fully factored form of the equation is: \((x -1)(x + 1)(x + 3) = 0\)

Step 3: Solve for x

Solve each factor for x, these are the values for which the original denominator equals zero and must be excluded from the domain: \(x -1 = 0\), \(x = 1\) \(x + 1 = 0\), \(x = -1\) \(x + 3 = 0\), \(x = -3\)