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Partial Fraction Decomposition of x⁴ - x² / [x²(x² - 1)²]

To resolve the expression into partial fractions, we need to find the unknown constants A, B, and C such that:

x⁴ - x² = A/x + B/x² + C/(x - 1) + D/(x + 1) + E/(x - 1)² + F/(x + 1)²

Step 1: Finding the Common Denominator
The given expression can be written as:
x⁴ - x² = [A(x² - 1)² + Bx(x² - 1)(x - 1)² + Cx²(x + 1)(x - 1)² + Dx²(x + 1)² + Ex(x + 1)(x - 1)² + F(x + 1)²]/[x²(x² - 1)²]

Step 2: Expanding the Numerator
Expanding the numerator, we get:
x⁴ - x² = [A(x⁴ - 2x² + 1) + Bx(x⁴ - 3x³ + 3x² - x) + Cx²(x⁴ - 2x² + 1) + Dx²(x⁴ + 2x² + 1) + Ex(x⁴ - 2x² + 1) + F(x⁴ + 2x² + 1)]/[x²(x² - 1)²]

Step 3: Combining Like Terms
Grouping the terms with similar powers of x, we have:
x⁴ - x² = [(A + C + D + F)x⁴ + (-2A + B - 3B + C - 2C + D + E)x³ + (A + 3B + C + D - 2E + F)x² + (-B + D + F)x + (A + F)]/[x²(x² - 1)²]

Step 4: Equating Coefficients
Equating the coefficients of each power of x on both sides, we get the following system of equations:

Coefficient of x⁴: A + C + D + F = 1
Coefficient of x³: -2A + B - 3B + C - 2C + D + E = 0
Coefficient of x²: A + 3B + C + D - 2E + F = 0
Coefficient of x: -B + D + F = 0
Coefficient of x⁰: A + F = 0

Step 5: Solving the System of Equations
Solving the system of equations, we find:
A = -1/2
B = 0
C = 1/2
D = 0
E = 1/2
F = 1/2

Step 6: Writing the Partial Fraction Decomposition
Substituting the values of A, B, C, D, E, and F into the partial fraction decomposition, we have:
x⁴ - x²/[x²(x² -