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
![**Dividing Rational Expressions**
To find the quotient, we will use the rule for dividing rational expressions. After multiplying by the reciprocal, we will factor each polynomial that is not prime and remove any common factors of the numerator and denominator.
We want to give the result in simplified form.
### Step-by-Step Solution:
Given expression:
\[ \frac{x^2 - 2x + 4}{4x + 4} \div \frac{x^2 - 2x + 4}{2x^2 - 2} \]
1. **Reciprocal of the Second Rational Expression:**
\[ \frac{x^2 - 2x + 4}{4x + 4} \times \frac{2x^2 - 2}{x^2 - 2x + 4} \]
2. **Multiplying the Numerators and Denominators:**
\[ \frac{(x^2 - 2x + 4)(2x^2 - 2)}{(4x + 4)(x^2 - 2x + 4)} \]
3. **Factoring Completely:**
\[ x^2 - 2x + 4 \text{ is prime. Then simplify.} \]
4. **Factoring the Other Terms:**
\[ \frac{(x-1)^2 \times (2(x^2-1))}{(4(x+1)) \times (x^2-2x+4)} = \frac{(x-1)^2 \times 2(x-1)}{(4(x+1)) \times (x^2-2x+4)} = \frac{2(x-1)^3}{4(x+1)(x^2-2x+4)} \]
5. **Removing Common Factors and Simplifying:**
\[ \frac{2(x-1)(x^2-2x+4)}{4(x+1)} = \frac{x-1}{2(x+1)} \]
So, the simplified form is:
\[ \boxed{\frac{x-1}{2(x+1)}} \]
### Practice Problem:
Divide and simplify:
\[ \frac{x^2 + 2x + 4}{49x - 49} \div \frac{x^2 + 2x + 4}{7x^2 - 7x} \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F758f6220-2c9c-498c-a499-60dbbe8461af%2F32c15818-fdf3-4380-a441-09991961cbaa%2Fgtcfgib_processed.png&w=3840&q=75)
Transcribed Image Text:**Dividing Rational Expressions**
To find the quotient, we will use the rule for dividing rational expressions. After multiplying by the reciprocal, we will factor each polynomial that is not prime and remove any common factors of the numerator and denominator.
We want to give the result in simplified form.
### Step-by-Step Solution:
Given expression:
\[ \frac{x^2 - 2x + 4}{4x + 4} \div \frac{x^2 - 2x + 4}{2x^2 - 2} \]
1. **Reciprocal of the Second Rational Expression:**
\[ \frac{x^2 - 2x + 4}{4x + 4} \times \frac{2x^2 - 2}{x^2 - 2x + 4} \]
2. **Multiplying the Numerators and Denominators:**
\[ \frac{(x^2 - 2x + 4)(2x^2 - 2)}{(4x + 4)(x^2 - 2x + 4)} \]
3. **Factoring Completely:**
\[ x^2 - 2x + 4 \text{ is prime. Then simplify.} \]
4. **Factoring the Other Terms:**
\[ \frac{(x-1)^2 \times (2(x^2-1))}{(4(x+1)) \times (x^2-2x+4)} = \frac{(x-1)^2 \times 2(x-1)}{(4(x+1)) \times (x^2-2x+4)} = \frac{2(x-1)^3}{4(x+1)(x^2-2x+4)} \]
5. **Removing Common Factors and Simplifying:**
\[ \frac{2(x-1)(x^2-2x+4)}{4(x+1)} = \frac{x-1}{2(x+1)} \]
So, the simplified form is:
\[ \boxed{\frac{x-1}{2(x+1)}} \]
### Practice Problem:
Divide and simplify:
\[ \frac{x^2 + 2x + 4}{49x - 49} \div \frac{x^2 + 2x + 4}{7x^2 - 7x} \]
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