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!

---

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.