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Chapter 0: Problem 3

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

Short Answer

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

Step by step solution

01

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 \(x^{2}-25=0\).
02

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 \(x=-5\).
03

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.

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Most popular questions from this chapter

Will help you prepare for the material covered in the next section.Exercises \(144-146\) will help you prepare for the material covered in the next section. Factor the numerator and the denominator. Then simplify by dividing out the common factor in the numerator and the denominator. $$ \frac{x^{2}+6 x+5}{x^{2}-25} $$

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