Find the partial fraction decomposition for each rational expression. See Examples 1–4. (4x^2 - x - 15)/(x(x + 1)(x - 1))

Verified step by step guidance

1. Identify the form of the partial fraction decomposition based on the factors in the denominator. Since the denominator is \(x(x + 1)(x - 1)\), which consists of three distinct linear factors, the decomposition will be of the form: \(\frac{A}{x} + \frac{B}{x + 1} + \frac{C}{x - 1}\), where \(A\), \(B\), and \(C\) are constants to be determined.
2. Write the equation equating the original rational expression to the sum of the partial fractions: \(\frac{4x^{2} - x - 15}{x(x + 1)(x - 1)} = \frac{A}{x} + \frac{B}{x + 1} + \frac{C}{x - 1}\).
3. Multiply both sides of the equation by the common denominator \(x(x + 1)(x - 1)\) to clear the denominators, resulting in: \$4x^{2} - x - 15 = A(x + 1)(x - 1) + B x (x - 1) + C x (x + 1)$.
4. Expand each term on the right-hand side: - \(A(x + 1)(x - 1) = A(x^{2} - 1)\), - \(B x (x - 1) = B(x^{2} - x)\), - \(C x (x + 1) = C(x^{2} + x)\). Then combine like terms to express the right side as a polynomial in \(x\).
5. Set the coefficients of corresponding powers of \(x\) on both sides equal to each other to form a system of equations. Solve this system for \(A\), \(B\), and \(C\) to find the constants for the partial fraction decomposition.

Key Concepts

Partial Fraction Decomposition

Partial fraction decomposition is a method used to express a complex rational expression as a sum of simpler fractions with linear or quadratic denominators. This technique simplifies integration and other algebraic operations by breaking down the original fraction into manageable parts.

Factoring Polynomials

Factoring involves rewriting a polynomial as a product of its factors. In this problem, the denominator is already factored into linear terms x, (x + 1), and (x - 1), which is essential for setting up the partial fractions correctly.

Setting Up and Solving Systems of Equations

After expressing the rational expression as a sum of partial fractions with unknown coefficients, you multiply both sides by the common denominator and equate coefficients of corresponding powers of x. This process leads to a system of linear equations that must be solved to find the unknown constants.