Problem 104 - Algebra and Trigonometry

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

Step 1: Write down the expression for which you want to find the exclusions

The given expression is \(\frac{x^{3}-2 x^{2}-9 x+18}{x^{3}+3 x^{2}-x-3}\).

Step 2: Set the denominator equal to zero

The denominator of the expression is \(x^{3}+3 x^{2}-x-3\). We need to find the values of \(x\) for which this is equal to zero, so we write the equation: \(x^{3}+3 x^{2}-x-3=0\).

Step 3: Solve the equation

Solving the equation will require you to use factoring, the quadratic formula, or other methods depending on the complexity of the denominator. However, for this cubic equation, one of the methods to solve it will be by factoring.By factoring, we find that \( (x-1)(x+1)(x+3) = 0 \).

Step 4: Find the values for \(x\)

Setting each factor equal to zero gives \(x=1\), \(x=-1\), and \(x=-3\). These are values of \(x\) that make the denominator equal to zero.