Write the partial fraction decomposition of each rational expression. 4x² - 7x - 3/(x³ -x)

Verified step by step guidance

1. First, factor the denominator \(x^3 - x\). Notice that you can factor out an \(x\) to get \(x(x^2 - 1)\).
2. Recognize that \(x^2 - 1\) is a difference of squares, so factor it further as \(x(x - 1)(x + 1)\).
3. Set up the partial fraction decomposition with unknown constants for each factor in the denominator: \(\frac{4x^2 - 7x - 3}{x(x - 1)(x + 1)} = \frac{A}{x} + \frac{B}{x - 1} + \frac{C}{x + 1}\).
4. Multiply both sides of the equation by the common denominator \(x(x - 1)(x + 1)\) to clear the fractions, resulting in: \$4x^2 - 7x - 3 = A(x - 1)(x + 1) + B x (x + 1) + C x (x - 1)$.
5. Expand the right side and collect like terms to form a polynomial equation. Then, equate the coefficients of corresponding powers of \(x\) on both sides to create a system of equations to solve for \(A\), \(B\), and \(C\).

Key Concepts

Partial Fraction Decomposition

Partial fraction decomposition is a method used to express a rational function as a sum of simpler fractions with denominators that are factors of the original denominator. This technique simplifies integration and other algebraic operations by breaking down complex fractions into manageable parts.

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of its factors. For partial fractions, factoring the denominator completely into linear and/or irreducible quadratic factors is essential to set up the correct form of the decomposition.

Setting Up and Solving Equations for Coefficients

After expressing the rational function as a sum of partial fractions, you equate the original numerator to the combined numerator of the decomposed fractions. This leads to a system of equations for the unknown coefficients, which you solve to find the values that complete the decomposition.