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,...
Related questions
Question
Rational expression
![**Perform the Division**
We have the following expression to solve:
\[
\frac{36 - z^2}{z^2 + 5z + 4} \div \frac{z^2 - 5z - 6}{z^2 + 4z}
\]
This expression is set up for division. To simplify, we need to multiply by the reciprocal of the second fraction.
### Steps to Solve:
1. **Identify the Complex Fractions**:
- Numerator 1: \(36 - z^2\)
- Denominator 1: \(z^2 + 5z + 4\)
- Numerator 2: \(z^2 - 5z - 6\)
- Denominator 2: \(z^2 + 4z\)
2. **Change to Multiplication**:
\[
\frac{36 - z^2}{z^2 + 5z + 4} \times \frac{z^2 + 4z}{z^2 - 5z - 6}
\]
3. **Factor Each Polynomial**:
- \(36 - z^2\) can be rewritten as \((6 - z)(6 + z)\).
- \(z^2 + 5z + 4\) can be factored as \((z + 1)(z + 4)\).
- \(z^2 - 5z - 6\) can be factored as \((z - 6)(z + 1)\).
- \(z^2 + 4z\) can be factored as \(z(z + 4)\).
4. **Rewrite the Expression**:
\[
\frac{(6 - z)(6 + z)}{(z + 1)(z + 4)} \times \frac{z(z + 4)}{(z - 6)(z + 1)}
\]
5. **Simplify by Canceling Common Factors**:
- Cancel \((z + 1)\) and \((z + 4)\) from numerator and denominator.
- Simplify to get:
\[
\frac{(6 - z)z}{(z - 6)}
\]
This is the simplified expression. If further simplification is needed or if the context](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6c693121-0c53-4461-96bf-03d77f2fd62b%2F3b9d2c54-fb25-4864-9bf9-f94710f65b66%2Fu4j4lwg_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Perform the Division**
We have the following expression to solve:
\[
\frac{36 - z^2}{z^2 + 5z + 4} \div \frac{z^2 - 5z - 6}{z^2 + 4z}
\]
This expression is set up for division. To simplify, we need to multiply by the reciprocal of the second fraction.
### Steps to Solve:
1. **Identify the Complex Fractions**:
- Numerator 1: \(36 - z^2\)
- Denominator 1: \(z^2 + 5z + 4\)
- Numerator 2: \(z^2 - 5z - 6\)
- Denominator 2: \(z^2 + 4z\)
2. **Change to Multiplication**:
\[
\frac{36 - z^2}{z^2 + 5z + 4} \times \frac{z^2 + 4z}{z^2 - 5z - 6}
\]
3. **Factor Each Polynomial**:
- \(36 - z^2\) can be rewritten as \((6 - z)(6 + z)\).
- \(z^2 + 5z + 4\) can be factored as \((z + 1)(z + 4)\).
- \(z^2 - 5z - 6\) can be factored as \((z - 6)(z + 1)\).
- \(z^2 + 4z\) can be factored as \(z(z + 4)\).
4. **Rewrite the Expression**:
\[
\frac{(6 - z)(6 + z)}{(z + 1)(z + 4)} \times \frac{z(z + 4)}{(z - 6)(z + 1)}
\]
5. **Simplify by Canceling Common Factors**:
- Cancel \((z + 1)\) and \((z + 4)\) from numerator and denominator.
- Simplify to get:
\[
\frac{(6 - z)z}{(z - 6)}
\]
This is the simplified expression. If further simplification is needed or if the context
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