https://file.aiquickdraw.com/mathbot/file/1725917389743npzb1a7u.jpg

The image contains three mathematical expressions that involve division of rational functions. Let's work through them step-by-step.

### (1) Expression:
\[
\frac{x^2 - 1}{x^2 - 4} \div \frac{x + 1}{x - 2}
\]
Steps:
1. Rewrite division as multiplication by the reciprocal:
   \[
   \frac{x^2 - 1}{x^2 - 4} \times \frac{x - 2}{x + 1}
   \]
2. Factor the numerators and denominators where possible:
   \[
   \frac{(x - 1)(x + 1)}{(x - 2)(x + 2)} \times \frac{x - 2}{x + 1}
   \]
3. Simplify by canceling out common terms:
   \[
   = \frac{x - 1}{x + 2}
   \]

### (2) Expression:
\[
\frac{x - 3}{x + 3} \div \frac{x + 2}{x^2 + 2x - 3}
\]
Steps:
1. Rewrite division as multiplication by the reciprocal:
   \[
   \frac{x - 3}{x + 3} \times \frac{x^2 + 2x - 3}{x + 2}
   \]
2. Factor the quadratic expression:
   \[
   \frac{x - 3}{x + 3} \times \frac{(x - 1)(x + 3)}{x + 2}
   \]
3. Simplify by canceling out common terms:
   \[
   = \frac{x - 3}{x + 2} \times (x - 1)
   \]
   Final result:
   \[
   = \frac{(x - 3)(x - 1)}{x + 2}
   \]

### (3) Expression:
\[
\frac{x + 1}{x^2 - 4x + 4} \div \frac{x}{x^2 - 3x + 2}
\]
Steps:
1. Rewrite division as multiplication by the reciprocal:
   \[
   \frac{x + 1}{(x - 2)^2} \times \frac{x^2 - 3x + 2}{x}
   \]
2. Factor the quadratic expressions:
   \[
   \frac{x + 1}{(x - 2)^2} \times \frac{(x - 1)(x - 2)}{x}
   \]
3. Simplify by canceling out common terms:
   \[
   = \frac{x + 1}{x(x - 2)}
   \]

Let me know if you would like more detailed explanations or further clarifications!

Here are five related questions to expand on this:

1. How do you identify which terms can be factored in a rational function?
2. What rules are used to simplify expressions involving division of fractions?
3. How can common factors in the numerator and denominator affect the domain of a rational function?
4. What are the steps for performing polynomial long division when factoring isn't straightforward?
5. How can multiplying by the reciprocal help in solving complex rational equations?

**Tip:** Always check for any values that make the denominator zero in rational functions, as these are restrictions on the domain.

Solve the following rational expressions: (1) (x^2 - 1)/(x^2 - 4) ÷ (x + 1)/(x - 2), (2) (x - 3)/(x + 3) ÷ (x + 2)/(x^2 + 2x - 3), (3) (x + 1)/(x^2 - 4x + 4) ÷ x/(x^2 - 3x + 2).

Learn how to solve rational expressions involving division of quadratic functions. This guide covers key algebraic concepts like factoring quadratics, simplifying fractions, and the factor theorem. Suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation