Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![### Algebra: Determining Excluded Values
#### Problem Statement:
Reduce the following expression and determine the excluded value(s) for \( x \):
\[ \frac{2x^2 - 8}{x^2 - 2x - 8} \]
#### Answer Choices:
- \( x \neq 0, 4 \)
- \( x \neq \pm 4 \)
- \( x \neq \pm 2 \)
- \( x \neq -2, 4 \)
In this problem, you'll need to simplify the given rational expression and then determine the values of \( x \) that make the denominator equal to zero. These values are called the excluded values because, for those values, the expression would be undefined.
Let's start with the simplification process:
1. **Factor the numerator and the denominator** wherever possible.
2. **Identify and exclude** the values of \( x \) that make the denominator zero.
#### Step-by-Step Solution:
1. **Factor the numerator:**
The numerator, \( 2x^2 - 8 \), can be factored as:
\[ 2(x^2 - 4) = 2(x - 2)(x + 2) \]
2. **Factor the denominator:**
The denominator, \( x^2 - 2x - 8 \), can be factored as:
\[ (x - 4)(x + 2) \]
Now the expression looks like:
\[ \frac{2(x - 2)(x + 2)}{(x - 4)(x + 2)} \]
After simplifying by cancelling out the common factors \((x + 2)\) we get:
\[ \frac{2(x - 2)}{(x - 4)} \]
3. **Determine the excluded values:**
To find the excluded values, we set the denominator equal to zero:
\[ x^2 - 2x - 8 = 0 \]
Factoring it:
\[ (x - 4)(x + 2) = 0 \]
So, \( x \) can be \( 4 \) or \(-2 \). These values are excluded because they make the denominator zero.
Therefore, \( x \neq -2, 4 \).
#### Correct Answer:
- \( x \neq -](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd66c9551-2207-4a1f-96cc-82c211723c9b%2F07905051-2eac-46e0-bab6-40e1215facb6%2Fbi2qvws_processed.png&w=3840&q=75)
Transcribed Image Text:### Algebra: Determining Excluded Values
#### Problem Statement:
Reduce the following expression and determine the excluded value(s) for \( x \):
\[ \frac{2x^2 - 8}{x^2 - 2x - 8} \]
#### Answer Choices:
- \( x \neq 0, 4 \)
- \( x \neq \pm 4 \)
- \( x \neq \pm 2 \)
- \( x \neq -2, 4 \)
In this problem, you'll need to simplify the given rational expression and then determine the values of \( x \) that make the denominator equal to zero. These values are called the excluded values because, for those values, the expression would be undefined.
Let's start with the simplification process:
1. **Factor the numerator and the denominator** wherever possible.
2. **Identify and exclude** the values of \( x \) that make the denominator zero.
#### Step-by-Step Solution:
1. **Factor the numerator:**
The numerator, \( 2x^2 - 8 \), can be factored as:
\[ 2(x^2 - 4) = 2(x - 2)(x + 2) \]
2. **Factor the denominator:**
The denominator, \( x^2 - 2x - 8 \), can be factored as:
\[ (x - 4)(x + 2) \]
Now the expression looks like:
\[ \frac{2(x - 2)(x + 2)}{(x - 4)(x + 2)} \]
After simplifying by cancelling out the common factors \((x + 2)\) we get:
\[ \frac{2(x - 2)}{(x - 4)} \]
3. **Determine the excluded values:**
To find the excluded values, we set the denominator equal to zero:
\[ x^2 - 2x - 8 = 0 \]
Factoring it:
\[ (x - 4)(x + 2) = 0 \]
So, \( x \) can be \( 4 \) or \(-2 \). These values are excluded because they make the denominator zero.
Therefore, \( x \neq -2, 4 \).
#### Correct Answer:
- \( x \neq -
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