Find the partial fraction decomposition for each rational expression. See Examples 1–4. (x²)/(x⁴ - 1)

Verified step by step guidance

1. Start by factoring the denominator \(x^4 - 1\). Recognize this as a difference of squares: \(x^4 - 1 = (x^2)^2 - 1^2 = (x^2 - 1)(x^2 + 1)\).
2. Further factor \(x^2 - 1\) as another difference of squares: \(x^2 - 1 = (x - 1)(x + 1)\). So the full factorization of the denominator is \(x^4 - 1 = (x - 1)(x + 1)(x^2 + 1)\).
3. Set up the partial fraction decomposition with unknown constants for each factor: \[\frac{x^2}{(x - 1)(x + 1)(x^2 + 1)} = \frac{A}{x - 1} + \frac{B}{x + 1} + \frac{Cx + D}{x^2 + 1}\] Note that for the quadratic factor \(x^2 + 1\), the numerator is linear (\(Cx + D\)).
4. Multiply both sides of the equation by the denominator \((x - 1)(x + 1)(x^2 + 1)\) to clear the fractions, resulting in: \[x^2 = A(x + 1)(x^2 + 1) + B(x - 1)(x^2 + 1) + (Cx + D)(x - 1)(x + 1)\]
5. Expand the right-hand side and collect like terms in powers of \(x\). Then, equate the coefficients of corresponding powers of \(x\) on both sides to form a system of equations. Solve this system to find the values of \(A\), \(B\), \(C\), and \(D\).

Key Concepts

Partial Fraction Decomposition

Partial fraction decomposition is a method used to express a complex rational expression as a sum of simpler fractions. This technique is especially useful for integrating rational functions or simplifying expressions. It involves breaking down the denominator into factors and assigning unknown constants to each fraction.

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of its factors. For partial fractions, factoring the denominator completely into linear or irreducible quadratic factors is essential. For example, x^4 - 1 factors as (x^2 - 1)(x^2 + 1), and further as (x - 1)(x + 1)(x^2 + 1).

Setting Up and Solving Systems of Equations

After expressing the rational function as a sum of partial fractions with unknown coefficients, you multiply both sides by the denominator to clear fractions. Then, equate coefficients of corresponding powers of x to form a system of linear equations. Solving this system finds the values of the unknown constants.