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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Question
![**Simplify the Given Expression:**
\[ \frac{a + 2}{a^2 + a - 2} + \frac{4}{a^2 + 2a - 3} \]
### Steps to Simplify:
1. **Factor the Denominators:**
- The denominator \(a^2 + a - 2\) can be factored as:
\[ a^2 + a - 2 = (a + 2)(a - 1) \]
- The denominator \(a^2 + 2a - 3\) can be factored as:
\[ a^2 + 2a - 3 = (a + 3)(a - 1) \]
2. **Rewrite the Original Expression with Factored Denominators:**
\[ \frac{a + 2}{(a + 2)(a - 1)} + \frac{4}{(a + 3)(a - 1)} \]
3. **Identify the Common Denominator:**
The common denominator for this expression is \((a+2)(a-1)(a+3)\).
4. **Rewrite Each Fraction with the Common Denominator:**
- For the first fraction:
\[ \frac{a + 2}{(a + 2)(a - 1)} = \frac{a + 2}{(a + 2)(a - 1)} \times \frac{a + 3}{a + 3} = \frac{(a+2)(a+3)}{(a + 2)(a - 1)(a + 3)} = \frac{a^2 + 5a + 6}{(a + 2)(a - 1)(a + 3)} \]
- For the second fraction:
\[ \frac{4}{(a + 3)(a - 1)} = \frac{4}{(a + 3)(a - 1)} \times \frac{a + 2}{a + 2} = \frac{4(a + 2)}{(a + 2)(a - 1)(a + 3)} = \frac{4a + 8}{(a + 2)(a - 1)(a + 3](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2d158991-3671-4459-aef6-bd0615668acc%2F621529bd-7eb6-4c87-8cf2-dcc6e36c341a%2Fhph961c_processed.png&w=3840&q=75)
Transcribed Image Text:**Simplify the Given Expression:**
\[ \frac{a + 2}{a^2 + a - 2} + \frac{4}{a^2 + 2a - 3} \]
### Steps to Simplify:
1. **Factor the Denominators:**
- The denominator \(a^2 + a - 2\) can be factored as:
\[ a^2 + a - 2 = (a + 2)(a - 1) \]
- The denominator \(a^2 + 2a - 3\) can be factored as:
\[ a^2 + 2a - 3 = (a + 3)(a - 1) \]
2. **Rewrite the Original Expression with Factored Denominators:**
\[ \frac{a + 2}{(a + 2)(a - 1)} + \frac{4}{(a + 3)(a - 1)} \]
3. **Identify the Common Denominator:**
The common denominator for this expression is \((a+2)(a-1)(a+3)\).
4. **Rewrite Each Fraction with the Common Denominator:**
- For the first fraction:
\[ \frac{a + 2}{(a + 2)(a - 1)} = \frac{a + 2}{(a + 2)(a - 1)} \times \frac{a + 3}{a + 3} = \frac{(a+2)(a+3)}{(a + 2)(a - 1)(a + 3)} = \frac{a^2 + 5a + 6}{(a + 2)(a - 1)(a + 3)} \]
- For the second fraction:
\[ \frac{4}{(a + 3)(a - 1)} = \frac{4}{(a + 3)(a - 1)} \times \frac{a + 2}{a + 2} = \frac{4(a + 2)}{(a + 2)(a - 1)(a + 3)} = \frac{4a + 8}{(a + 2)(a - 1)(a + 3
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