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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![### Simplifying Rational Expressions
To simplify the given expression, follow these steps:
1. **Expression Provided**:
\[
\frac{8p^2 - 71pq - 9q^2}{p^2 - 81q^2} \div \frac{8p^2 + 65pq + 8q^2}{8p^2 + 71pq - 9q^2}
\]
2. **Step-by-Step Solution**:
Let's first rewrite the division as a multiplication by the reciprocal.
\[
\frac{8p^2 - 71pq - 9q^2}{p^2 - 81q^2} \times \frac{8p^2 + 71pq - 9q^2}{8p^2 + 65pq + 8q^2}
\]
3. **Factoring Each Polynomial**:
Factor each polynomial where possible.
- **Numerator for first fraction**: \( 8p^2 - 71pq - 9q^2 \) (This needs factoring)
- **Denominator for first fraction**: \( p^2 - 81q^2 \) is a difference of squares:
\[
p^2 - 81q^2 = (p - 9q)(p + 9q)
\]
- **Numerator for second fraction**: \( 8p^2 + 71pq - 9q^2 \) (This needs factoring)
- **Denominator for second fraction**: \( 8p^2 + 65pq + 8q^2 \) (This needs factoring)
4. **Combining**:
Once each polynomial is factored, rewrite the expression, cancel common factors, and simplify.
5. **Simplify**:
The simplified form will be achieved by canceling out the common terms in the numerator and denominator.
This step-by-step method allows for the systematic simplification of the rational expression. Once all polynomials are factored correctly, the simplification becomes straightforward.
- Ensure accuracy in factoring.
- Cancel only like terms in the fraction’s numerator and denominator.
Use these guidelines to simplify similar rational expressions generally.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa7d8b63c-f6b2-4537-9353-75fc012a2847%2F986cf6cf-ebf6-4ed7-bfab-8c917b34be7c%2F2b9q0mu_processed.png&w=3840&q=75)
Transcribed Image Text:### Simplifying Rational Expressions
To simplify the given expression, follow these steps:
1. **Expression Provided**:
\[
\frac{8p^2 - 71pq - 9q^2}{p^2 - 81q^2} \div \frac{8p^2 + 65pq + 8q^2}{8p^2 + 71pq - 9q^2}
\]
2. **Step-by-Step Solution**:
Let's first rewrite the division as a multiplication by the reciprocal.
\[
\frac{8p^2 - 71pq - 9q^2}{p^2 - 81q^2} \times \frac{8p^2 + 71pq - 9q^2}{8p^2 + 65pq + 8q^2}
\]
3. **Factoring Each Polynomial**:
Factor each polynomial where possible.
- **Numerator for first fraction**: \( 8p^2 - 71pq - 9q^2 \) (This needs factoring)
- **Denominator for first fraction**: \( p^2 - 81q^2 \) is a difference of squares:
\[
p^2 - 81q^2 = (p - 9q)(p + 9q)
\]
- **Numerator for second fraction**: \( 8p^2 + 71pq - 9q^2 \) (This needs factoring)
- **Denominator for second fraction**: \( 8p^2 + 65pq + 8q^2 \) (This needs factoring)
4. **Combining**:
Once each polynomial is factored, rewrite the expression, cancel common factors, and simplify.
5. **Simplify**:
The simplified form will be achieved by canceling out the common terms in the numerator and denominator.
This step-by-step method allows for the systematic simplification of the rational expression. Once all polynomials are factored correctly, the simplification becomes straightforward.
- Ensure accuracy in factoring.
- Cancel only like terms in the fraction’s numerator and denominator.
Use these guidelines to simplify similar rational expressions generally.
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