In Exercises 9–42, write the partial fraction decomposition of each rational expression. 4x²+13x-9/x (x − 1)(x+3)

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1. Identify the rational expression: \( \frac{4x^2 + 13x - 9}{x (x - 1)(x + 3)} \). Notice that the denominator is factored into linear factors \( x \), \( x - 1 \), and \( x + 3 \).
2. Set up the partial fraction decomposition by expressing the rational expression as a sum of fractions with unknown constants in the numerators: \( \frac{A}{x} + \frac{B}{x - 1} + \frac{C}{x + 3} \), where \( A \), \( B \), and \( C \) are constants to be determined.
3. Multiply both sides of the equation by the common denominator \( x (x - 1)(x + 3) \) to clear the denominators: \( 4x^2 + 13x - 9 = A (x - 1)(x + 3) + B x (x + 3) + C x (x - 1) \).
4. Expand the right-hand side by multiplying out each term: \( A (x^2 + 2x - 3) + B (x^2 + 3x) + C (x^2 - x) \). Then combine like terms to write the right side as a quadratic expression in terms of \( x^2 \), \( x \), and the constant term.
5. Equate the coefficients of corresponding powers of \( x \) from both sides of the equation to form a system of linear equations in \( A \), \( B \), and \( C \). Solve this system to find the values of \( A \), \( B \), and \( C \).

Key Concepts

Rational Expressions

A rational expression is a fraction where both the numerator and denominator are polynomials. Understanding how to manipulate these expressions, including factoring and simplifying, is essential before performing partial fraction decomposition.

Partial Fraction Decomposition

Partial fraction decomposition involves expressing a complex rational expression as a sum of simpler fractions with linear or quadratic denominators. This technique is useful for integration and solving equations involving rational expressions.

Factoring Polynomials

Factoring polynomials means rewriting a polynomial as a product of its factors. Recognizing and factoring the denominator into linear factors, such as (x)(x−1)(x+3), is crucial for setting up the correct form of partial fractions.