Problem 3 - College Algebra Essentials

Question

Find all numbers that must be excluded from the domain of each rational expression. $$\frac{x+5}{x^{2}-25}$$

Step-by-Step Solution

Answer

The numbers that must be excluded from the domain of the given rational expression are \(x = -5\) and \(x = 5\).

Step 1: Identify the denominator

Firstly, identify the denominator of the fraction which is \(x^{2}-25\)

Step 2: Set the denominator equal to zero

In order to find the values making this zero, set the denominator equal to zero and solve: \(x^{2}-25 = 0\)

Step 3: Solve the equation

This equation can be factored using the difference of squares: \(x^{2}-5^{2} = 0\), so \((x+5)(x-5) = 0\). Setting each factor equal to zero and solving for \(x\), we get \(x = -5\) and \(x = 5\).

Step 4: Identify the numbers to be excluded

The numbers that must be excluded from the domain are therefore \(x = -5\) and \(x = 5\), because these values would make the denominator equal to zero.