Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN: 9780134463216
Author: Robert F. Blitzer
Publisher: PEARSON
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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
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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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