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
![**Multiplying and Dividing Polynomials**
**Problem 4:**
\[
\frac{4x + 12}{x + 3} \div \frac{x^2 + x - 6}{x - 2}
\]
- **Concept:**
The problem involves dividing two rational functions.
- **Explanation:**
To divide rational expressions, multiply by the reciprocal of the divisor. This means you'll take the second fraction, flip it, and then multiply.
- **Steps:**
1. **Factor where possible:**
- Numerator of the first fraction: \(4x + 12\) can be factored as \(4(x + 3)\).
- Denominator of the second fraction: \(x^2 + x - 6\) factors into \((x + 3)(x - 2)\).
2. **Rewrite the division as multiplication:**
\[
\frac{4(x + 3)}{x + 3} \times \frac{x - 2}{(x + 3)(x - 2)}
\]
3. **Cancel common terms:**
- \(x + 3\) cancels out in the first fraction.
- \(x - 2\) cancels out across.
4. **Simplify the expression:**
- You’ll be left with \(4\).
This procedure allows you to simplify complex rational expressions by breaking them into more manageable parts and canceling out terms effectively.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc37dbff0-e053-476c-a940-a199db5d9106%2Fe63ddb01-48e3-41bf-8895-24fc073cd728%2F8qbbl7_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Multiplying and Dividing Polynomials**
**Problem 4:**
\[
\frac{4x + 12}{x + 3} \div \frac{x^2 + x - 6}{x - 2}
\]
- **Concept:**
The problem involves dividing two rational functions.
- **Explanation:**
To divide rational expressions, multiply by the reciprocal of the divisor. This means you'll take the second fraction, flip it, and then multiply.
- **Steps:**
1. **Factor where possible:**
- Numerator of the first fraction: \(4x + 12\) can be factored as \(4(x + 3)\).
- Denominator of the second fraction: \(x^2 + x - 6\) factors into \((x + 3)(x - 2)\).
2. **Rewrite the division as multiplication:**
\[
\frac{4(x + 3)}{x + 3} \times \frac{x - 2}{(x + 3)(x - 2)}
\]
3. **Cancel common terms:**
- \(x + 3\) cancels out in the first fraction.
- \(x - 2\) cancels out across.
4. **Simplify the expression:**
- You’ll be left with \(4\).
This procedure allows you to simplify complex rational expressions by breaking them into more manageable parts and canceling out terms effectively.
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