Problem 3 - College Algebra

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 rational expression are \(x=5\) and \(x=-5\).

Step 1: Set denominator equal to zero

To find the numbers that must be excluded from the domain, set the denominator equal to zero and solve for \(x\). The denominator is \(x^{2}-25\), so the equation to solve is \(x^{2}-25=0\).

Step 2: Find roots of the equation

The given equation \(x^{2}-25=0\) is a difference of squares, so it can be factored as \((x-5)(x+5)=0\). Setting each factor equal to zero gives \(x-5=0\) or \(x+5=0\). Solving these equations for \(x\) gives \(x=5\) or \(x=-5\).

Step 3: Identify the numbers to exclude from the domain

The solutions to the equation \(x^{2}-25=0\) are \(x=5\) and \(x=-5\). These are the values that make the denominator of the rational function equal to zero, so these numbers must be excluded from the domain.