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
Find the quotient
![**Find the Quotient**
Given the expression:
\[
\frac{2}{(x+1)(x+3)^2} \div \frac{x^2-4}{x^2+4x+3}
\]
Simplify as much as possible.
To simplify, follow these steps:
1. **Rewrite the Division as Multiplication:**
\[
\frac{2}{(x+1)(x+3)^2} \times \frac{x^2+4x+3}{x^2-4}
\]
2. **Factor the Polynomials:**
- \(x^2 - 4\) factors to \((x-2)(x+2)\).
- \(x^2 + 4x + 3\) factors to \((x+1)(x+3)\).
3. **Substitute the Factored Forms:**
\[
\frac{2}{(x+1)(x+3)^2} \times \frac{(x+1)(x+3)}{(x-2)(x+2)}
\]
4. **Cancel Common Factors:**
- Cancel \((x+1)\) and one \((x+3)\) from the numerator and denominator.
5. **Remaining Expression:**
\[
\frac{2}{(x+3)(x+2)(x-2)}
\]
The result is:
\[
\frac{2}{(x+3)(x+2)(x-2)}
\]
This is the simplified form of the expression.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb88d9b02-73da-424d-af4e-50ad976cc081%2Fa587db15-a452-4865-b155-f345eb00c317%2Fms8p0d_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Find the Quotient**
Given the expression:
\[
\frac{2}{(x+1)(x+3)^2} \div \frac{x^2-4}{x^2+4x+3}
\]
Simplify as much as possible.
To simplify, follow these steps:
1. **Rewrite the Division as Multiplication:**
\[
\frac{2}{(x+1)(x+3)^2} \times \frac{x^2+4x+3}{x^2-4}
\]
2. **Factor the Polynomials:**
- \(x^2 - 4\) factors to \((x-2)(x+2)\).
- \(x^2 + 4x + 3\) factors to \((x+1)(x+3)\).
3. **Substitute the Factored Forms:**
\[
\frac{2}{(x+1)(x+3)^2} \times \frac{(x+1)(x+3)}{(x-2)(x+2)}
\]
4. **Cancel Common Factors:**
- Cancel \((x+1)\) and one \((x+3)\) from the numerator and denominator.
5. **Remaining Expression:**
\[
\frac{2}{(x+3)(x+2)(x-2)}
\]
The result is:
\[
\frac{2}{(x+3)(x+2)(x-2)}
\]
This is the simplified form of the expression.
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