Partial fraction decomposition of x²/x⁴-x²-2

To perform the partial fraction decomposition of the rational expression \( \frac{x^2}{x^4 - x^2 - 2} \), we first factor the denominator.

### Step 1: Factor the Denominator
The denominator is \( x^4 - x^2 - 2 \). We rewrite this as a quadratic in \( x^2 \):

\[
x^4 - x^2 - 2 = (x^2)^2 - x^2 - 2
\]

Now, we solve the quadratic equation by factoring or using the quadratic formula. Let's first try factoring it directly:

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

Thus, the factored form of the denominator is:

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

### Step 2: Set up the Partial Fractions
We now express \( \frac{x^2}{(x^2 - 2)(x^2 + 1)} \) as a sum of partial fractions:

\[
\frac{x^2}{(x^2 - 2)(x^2 + 1)} = \frac{Ax + B}{x^2 - 2} + \frac{Cx + D}{x^2 + 1}
\]

### Step 3: Clear the Denominator
Multiply both sides by \( (x^2 - 2)(x^2 + 1) \) to clear the denominator:

\[
x^2 = (Ax + B)(x^2 + 1) + (Cx + D)(x^2 - 2)
\]

### Step 4: Expand the Right-Hand Side
Now, expand both terms:

\[
(Ax + B)(x^2 + 1) = Ax^3 + Ax + Bx^2 + B
\]
\[
(Cx + D)(x^2 - 2) = Cx^3 - 2Cx + Dx^2 - 2D
\]

Adding both parts together:

\[
Ax^3 + Ax + Bx^2 + B + Cx^3 - 2Cx + Dx^2 - 2D
\]

Combine like terms:

\[
(A + C)x^3 + (B + D)x^2 + (A - 2C)x + (B - 2D)
\]

### Step 5: Set up a System of Equations
Now, equate the coefficients of the powers of \( x \) on both sides:

- Coefficient of \( x^3 \): \( A + C = 0 \)
- Coefficient of \( x^2 \): \( B + D = 1 \)
- Coefficient of \( x \): \( A - 2C = 0 \)
- Constant term: \( B - 2D = 0 \)

### Step 6: Solve the System of Equations

From \( A + C = 0 \), we have \( A = -C \).  
From \( A - 2C = 0 \), we have \( A = 2C \). Substituting \( A = -C \) into this, we get \( -C = 2C \), so \( C = 0 \) and \( A = 0 \).

Now, from \( B + D = 1 \) and \( B - 2D = 0 \), we can solve for \( B \) and \( D \). From \( B - 2D = 0 \), we have \( B = 2D \). Substituting this into \( B + D = 1 \), we get \( 2D + D = 1 \), so \( 3D = 1 \) and \( D = \frac{1}{3} \). Therefore, \( B = 2D = \frac{2}{3} \).

### Step 7: Write the Decomposition
Substituting the values of \( A \), \( B \), \( C \), and \( D \) into the partial fraction decomposition:

\[
\frac{x^2}{(x^2 - 2)(x^2 + 1)} = \frac{\frac{2}{3}}{x^2 - 2} + \frac{\frac{1}{3}}{x^2 + 1}
\]

Thus, the partial fraction decomposition is:

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

Let me know if you'd like further details!

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**Related Questions:**
1. How do you factor quartic polynomials?
2. Can you perform partial fraction decomposition with repeated factors?
3. How do you solve systems of equations from partial fractions?
4. What are the applications of partial fraction decomposition in calculus?
5. How can you handle irreducible quadratic factors in partial fractions?

**Tip:** Always start by factoring the denominator completely before setting up partial fractions.

This article explains how to perform partial fraction decomposition on the rational expression x²/(x⁴ - x² - 2). It covers key steps like factoring the polynomial, solving systems of equations, and applying the partial fraction formula. Suitable for Grades 10-12 students.

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