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
![### Polynomial Multiplication and Simplification
#### Problem Statement:
**Objective:** Multiply and simplify the given rational expressions, if possible.
\[
\frac{x^2 + 10x + 25}{x^2 + 11x + 30} \cdot \frac{x^2 + x - 30}{25 - x^2}
\]
#### Approach:
1. **Factorize each Polynomial**:
- **Numerator of the first fraction:** \( x^2 + 10x + 25 \)
- **Denominator of the first fraction:** \( x^2 + 11x + 30 \)
- **Numerator of the second fraction:** \( x^2 + x - 30 \)
- **Denominator of the second fraction:** \( 25 - x^2 \)
2. **Simplify by multiplying the fractions**:
- Express the given fractions in factored form if possible.
- Multiply the numerators together and the denominators together.
- Simplify, if possible, by canceling out common factors in the numerator and the denominator.
#### Step-by-Step Solution:
##### 1. Factorize the polynomials (if possible):
- **\( x^2 + 10x + 25 \)**:
\[
= (x + 5)(x + 5) = (x + 5)^2
\]
- **\( x^2 + 11x + 30 \)**:
\[
= (x + 5)(x + 6)
\]
- **\( x^2 + x - 30 \)**:
\[
= (x + 6)(x - 5)
\]
- **\( 25 - x^2 \)**:
\[
= (5 + x)(5 - x) = -(x - 5)(x + 5)
\]
##### 2. Substitute these factorizations into the fractions and combine:
\[
\frac{(x + 5)^2}{(x + 5)(x + 6)} \cdot \frac{(x + 6)(x - 5)}{-(x - 5)(x + 5)}
\]
##### 3. Simplify the expression by canceling out common factors:
- Cancel \((x + 5)\) and \((](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa7d8b63c-f6b2-4537-9353-75fc012a2847%2F73bfac2e-410f-442b-a366-beca21fcfac2%2Fuporf2l_processed.png&w=3840&q=75)
Transcribed Image Text:### Polynomial Multiplication and Simplification
#### Problem Statement:
**Objective:** Multiply and simplify the given rational expressions, if possible.
\[
\frac{x^2 + 10x + 25}{x^2 + 11x + 30} \cdot \frac{x^2 + x - 30}{25 - x^2}
\]
#### Approach:
1. **Factorize each Polynomial**:
- **Numerator of the first fraction:** \( x^2 + 10x + 25 \)
- **Denominator of the first fraction:** \( x^2 + 11x + 30 \)
- **Numerator of the second fraction:** \( x^2 + x - 30 \)
- **Denominator of the second fraction:** \( 25 - x^2 \)
2. **Simplify by multiplying the fractions**:
- Express the given fractions in factored form if possible.
- Multiply the numerators together and the denominators together.
- Simplify, if possible, by canceling out common factors in the numerator and the denominator.
#### Step-by-Step Solution:
##### 1. Factorize the polynomials (if possible):
- **\( x^2 + 10x + 25 \)**:
\[
= (x + 5)(x + 5) = (x + 5)^2
\]
- **\( x^2 + 11x + 30 \)**:
\[
= (x + 5)(x + 6)
\]
- **\( x^2 + x - 30 \)**:
\[
= (x + 6)(x - 5)
\]
- **\( 25 - x^2 \)**:
\[
= (5 + x)(5 - x) = -(x - 5)(x + 5)
\]
##### 2. Substitute these factorizations into the fractions and combine:
\[
\frac{(x + 5)^2}{(x + 5)(x + 6)} \cdot \frac{(x + 6)(x - 5)}{-(x - 5)(x + 5)}
\]
##### 3. Simplify the expression by canceling out common factors:
- Cancel \((x + 5)\) and \((
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