https://file.aiquickdraw.com/mathbot/file/17266688371506evdxk7r.jpg

The image shows a complex rational expression involving division of two fractions. The given expression is:

\[
\frac{3x^2 - 4x + 1}{x^2 - 16} \div \frac{2x^2 - x - 3}{x^2 + 4x + 4}
\]

### Steps to simplify:

1. **Division of Fractions Rule**: To divide fractions, we multiply the first fraction by the reciprocal of the second fraction. So, we rewrite the expression as:

\[
\frac{3x^2 - 4x + 1}{x^2 - 16} \times \frac{x^2 + 4x + 4}{2x^2 - x - 3}
\]

2. **Factor the Numerators and Denominators**: 
   - Factor \( x^2 - 16 \) (difference of squares): 
     \[
     x^2 - 16 = (x - 4)(x + 4)
     \]
   - Factor \( x^2 + 4x + 4 \) (perfect square trinomial):
     \[
     x^2 + 4x + 4 = (x + 2)(x + 2)
     \]
   - Factor \( 2x^2 - x - 3 \): Find two numbers whose product is \(-6\) and sum is \(-1\). The factorization is:
     \[
     2x^2 - x - 3 = (2x + 3)(x - 1)
     \]

3. **Multiply the Factored Expressions**:

   Now we have:
   \[
   \frac{3x^2 - 4x + 1}{(x - 4)(x + 4)} \times \frac{(x + 2)(x + 2)}{(2x + 3)(x - 1)}
   \]

   Next, check if the numerator \( 3x^2 - 4x + 1 \) can be factored further. In this case, it cannot be factored nicely.

4. **Simplify the Expression**:

   After multiplying, the expression becomes:
   \[
   \frac{(3x^2 - 4x + 1)(x + 2)(x + 2)}{(x - 4)(x + 4)(2x + 3)(x - 1)}
   \]

This is the simplified form of the original expression.

Would you like further clarification on any step or details?

### Related Questions:
1. How do you find the reciprocal of a rational expression?
2. What are the steps to factor quadratic expressions?
3. Why do we multiply by the reciprocal when dividing fractions?
4. Can you further simplify the expression if \(x = 4\) or \(x = -4\)?
5. How can you solve rational equations?

**Tip:** Always check for common factors to cancel out when simplifying rational expressions.

Simplify the rational expression: (3x^2 - 4x + 1) / (x^2 - 16) ÷ (2x^2 - x - 3) / (x^2 + 4x + 4)

Learn how to simplify the rational expression (3x^2 - 4x + 1) / (x^2 - 16) ÷ (2x^2 - x - 3) / (x^2 + 4x + 4) by using factoring and division of fractions. This detailed step-by-step solution is ideal for high school algebra students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation