\frac{1}{\left(x^2-1\right)^4}\left(\frac{6x^2-5}{1}-\frac{8x^3-20x+4}{x^2-1}\right)

To simplify the given expression:

\[
\frac{1}{(x^2 - 1)^4} \left(\frac{6x^2 - 5}{1} - \frac{8x^3 - 20x + 4}{x^2 - 1}\right)
\]

### Step 1: Simplify the expression inside the parentheses.

The expression inside the parentheses is:

\[
\frac{6x^2 - 5}{1} - \frac{8x^3 - 20x + 4}{x^2 - 1}
\]

This simplifies to:

\[
6x^2 - 5 - \frac{8x^3 - 20x + 4}{x^2 - 1}
\]

### Step 2: Combine the terms by getting a common denominator.

The common denominator is \( x^2 - 1 \). So, rewrite the first term with the common denominator:

\[
\frac{(6x^2 - 5)(x^2 - 1)}{x^2 - 1} - \frac{8x^3 - 20x + 4}{x^2 - 1}
\]

This becomes:

\[
\frac{(6x^2 - 5)(x^2 - 1) - (8x^3 - 20x + 4)}{x^2 - 1}
\]

### Step 3: Expand and simplify the numerator.

Expand \( (6x^2 - 5)(x^2 - 1) \):

\[
(6x^2 - 5)(x^2 - 1) = 6x^4 - 6x^2 - 5x^2 + 5 = 6x^4 - 11x^2 + 5
\]

Now, substitute it back into the expression:

\[
\frac{6x^4 - 11x^2 + 5 - (8x^3 - 20x + 4)}{x^2 - 1}
\]

Simplify the numerator:

\[
6x^4 - 8x^3 - 11x^2 + 20x + 1
\]

### Step 4: Substitute back into the original expression.

Now substitute this back into the original expression:

\[
\frac{1}{(x^2 - 1)^4} \cdot \frac{6x^4 - 8x^3 - 11x^2 + 20x + 1}{x^2 - 1}
\]

This simplifies to:

\[
\frac{6x^4 - 8x^3 - 11x^2 + 20x + 1}{(x^2 - 1)^5}
\]

So the simplified expression is:

\[
\frac{6x^4 - 8x^3 - 11x^2 + 20x + 1}{(x^2 - 1)^5}
\]

Would you like more details on any specific step, or do you have any further questions?

Here are some related questions you might consider:
1. How do you factor a higher degree polynomial like \(6x^4 - 8x^3 - 11x^2 + 20x + 1\)?
2. What are the properties of the function \(\frac{1}{(x^2-1)^4}\) as \(x\) approaches 1?
3. How can you find the roots of the polynomial in the numerator?
4. How do you find the derivative of a fraction with polynomials in both the numerator and the denominator?
5. How do you integrate a fraction of polynomials like the one obtained in the simplification?

**Tip:** Always look for common factors in both the numerator and the denominator when simplifying rational expressions.

This page provides a detailed step-by-step solution for simplifying the rational expression \frac{1}{\left(x^2-1\right)^4}\left(\frac{6x^2-5}{1}-\frac{8x^3-20x+4}{x^2-1}\right). Key concepts include algebraic simplification, polynomial division, and the common denominator method. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation