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
![The mathematical problem presented is as follows:
10. \[\frac{x + 3}{x^2 + 4x + 3} + \frac{x + 2}{x + 1}\]
In this expression, two rational functions are being added together. The first function's numerator is \(x + 3\) and its denominator is a quadratic polynomial \(x^2 + 4x + 3\). The second function's numerator is \(x + 2\) and its denominator is a linear polynomial \(x + 1\).
### Steps to Solve
1. **Factor the quadratic polynomial \(x^2 + 4x + 3\):**
\[
x^2 + 4x + 3 = (x + 1)(x + 3)
\]
2. **Rewrite the first fraction using the factored form of the denominator:**
\[
\frac{x + 3}{(x + 1)(x + 3)}
\]
3. **Simplify the first fraction:**
\[
\frac{x + 3}{(x + 1)(x + 3)} = \frac{1}{x + 1} \quad \text{for} \quad x \neq -3
\]
4. **Now substitute the simplified form of the first fraction into the original equation:**
\[
\frac{1}{x + 1} + \frac{x + 2}{x + 1}
\]
5. **Combine the fractions over a common denominator:**
\[
\frac{1 + (x + 2)}{x + 1} = \frac{x + 3}{x + 1}
\]
### Final Result
The expression simplifies to:
\[
\frac{x + 3}{x + 1}
\]
This is the solution to the given problem.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F40f3ec95-698a-4ff4-a856-55b7fb5ebe1e%2Ff8bb3fb2-7361-42ba-829d-1bf159fdac8c%2Fpy0ab7l_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The mathematical problem presented is as follows:
10. \[\frac{x + 3}{x^2 + 4x + 3} + \frac{x + 2}{x + 1}\]
In this expression, two rational functions are being added together. The first function's numerator is \(x + 3\) and its denominator is a quadratic polynomial \(x^2 + 4x + 3\). The second function's numerator is \(x + 2\) and its denominator is a linear polynomial \(x + 1\).
### Steps to Solve
1. **Factor the quadratic polynomial \(x^2 + 4x + 3\):**
\[
x^2 + 4x + 3 = (x + 1)(x + 3)
\]
2. **Rewrite the first fraction using the factored form of the denominator:**
\[
\frac{x + 3}{(x + 1)(x + 3)}
\]
3. **Simplify the first fraction:**
\[
\frac{x + 3}{(x + 1)(x + 3)} = \frac{1}{x + 1} \quad \text{for} \quad x \neq -3
\]
4. **Now substitute the simplified form of the first fraction into the original equation:**
\[
\frac{1}{x + 1} + \frac{x + 2}{x + 1}
\]
5. **Combine the fractions over a common denominator:**
\[
\frac{1 + (x + 2)}{x + 1} = \frac{x + 3}{x + 1}
\]
### Final Result
The expression simplifies to:
\[
\frac{x + 3}{x + 1}
\]
This is the solution to the given problem.
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