https://file.aiquickdraw.com/mathbot/file/1726419842207zzcawx8d.jpg

The problem asks to simplify the expression:

\[
\frac{y^2 + 2y}{y^2 - 4}
\]

and express the result in the form \(\frac{f(y)}{g(y)}\).

### Step 1: Factorize the numerator and the denominator
- The numerator \(y^2 + 2y\) can be factored as:
\[
y^2 + 2y = y(y + 2)
\]

- The denominator \(y^2 - 4\) is a difference of squares, which can be factored as:
\[
y^2 - 4 = (y - 2)(y + 2)
\]

### Step 2: Simplify the fraction
The expression becomes:

\[
\frac{y(y + 2)}{(y - 2)(y + 2)}
\]

Since \(y + 2\) appears in both the numerator and the denominator, we can cancel it (as long as \(y \neq -2\)):

\[
\frac{y}{y - 2}
\]

### Step 3: Identify \(f(y)\) and \(g(y)\)
From the simplified expression:

- \(f(y) = y\)
- \(g(y) = y - 2\)

### Final Answer:
- \(f(y) = y\)
- \(g(y) = y - 2\)

Would you like further clarification or details? 

Here are 5 related questions you might find useful:
1. How do you factor polynomials with higher degrees?
2. What is the general rule for factoring differences of squares?
3. How do you handle cases where the variable might make the denominator zero?
4. What are the steps to check the validity of simplifications in rational expressions?
5. How can we extend these concepts to more complex rational expressions?

**Tip:** Always check for restrictions in rational expressions where the denominator cannot be zero.

Simplify the expression (y^2 + 2y) / (y^2 - 4) and give your answer in the form of f(y) / g(y).

Learn how to simplify the rational expression (y^2 + 2y) / (y^2 - 4) using factoring techniques, including difference of squares. This step-by-step guide covers key algebraic concepts and is 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