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 Expression
Simplify the following mathematical expression. Show all steps to receive credit for your answer.
\[
\frac{3}{x^2 + 5x + 6} + \frac{2}{x + 3}
\]
### Steps:
1. **Factorize the Polynomial**:
- First, factorize \(x^2 + 5x + 6\):
\[ x^2 + 5x + 6 = (x + 2)(x + 3) \]
2. **Rewrite the Expression**:
- Rewrite the given expression using the factorized form:
\[ \frac{3}{(x + 2)(x + 3)} + \frac{2}{x + 3} \]
3. **Find a Common Denominator**:
- The common denominator for both fractions is \((x + 2)(x + 3)\):
\[ \frac{3}{(x + 2)(x + 3)} + \frac{2(x + 2)}{(x + 2)(x + 3)} \]
4. **Combine the Fractions**:
- Combine the two fractions over the common denominator:
\[ \frac{3 + 2(x + 2)}{(x + 2)(x + 3)} \]
5. **Simplify the Numerator**:
- Simplify the expression in the numerator:
\[ 3 + 2(x + 2) = 3 + 2x + 4 = 2x + 7 \]
6. **Write the Final Simplified Expression**:
- The simplified expression is:
\[ \frac{2x + 7}{(x + 2)(x + 3)} \]
### Final Answer:
\[
\frac{2x + 7}{(x + 2)(x + 3)}
\]
Use the text editor to follow these steps and show your work in detail. If you have an additional file or need to record a video explanation, use the options provided to add a file or record a video.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd66c9551-2207-4a1f-96cc-82c211723c9b%2Fc9f930ba-a84e-421b-919b-e675ab94d315%2Fhd98jy_processed.png&w=3840&q=75)
Transcribed Image Text:## Simplify the Expression
Simplify the following mathematical expression. Show all steps to receive credit for your answer.
\[
\frac{3}{x^2 + 5x + 6} + \frac{2}{x + 3}
\]
### Steps:
1. **Factorize the Polynomial**:
- First, factorize \(x^2 + 5x + 6\):
\[ x^2 + 5x + 6 = (x + 2)(x + 3) \]
2. **Rewrite the Expression**:
- Rewrite the given expression using the factorized form:
\[ \frac{3}{(x + 2)(x + 3)} + \frac{2}{x + 3} \]
3. **Find a Common Denominator**:
- The common denominator for both fractions is \((x + 2)(x + 3)\):
\[ \frac{3}{(x + 2)(x + 3)} + \frac{2(x + 2)}{(x + 2)(x + 3)} \]
4. **Combine the Fractions**:
- Combine the two fractions over the common denominator:
\[ \frac{3 + 2(x + 2)}{(x + 2)(x + 3)} \]
5. **Simplify the Numerator**:
- Simplify the expression in the numerator:
\[ 3 + 2(x + 2) = 3 + 2x + 4 = 2x + 7 \]
6. **Write the Final Simplified Expression**:
- The simplified expression is:
\[ \frac{2x + 7}{(x + 2)(x + 3)} \]
### Final Answer:
\[
\frac{2x + 7}{(x + 2)(x + 3)}
\]
Use the text editor to follow these steps and show your work in detail. If you have an additional file or need to record a video explanation, use the options provided to add a file or record a video.
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