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

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1. Identify the rational expression to decompose: \( \frac{x+4}{x^2 (x^2 + 4)} \). Notice the denominator is factored into \( x^2 \) and \( x^2 + 4 \), where \( x^2 \) is a repeated linear factor and \( x^2 + 4 \) is an irreducible quadratic factor.
2. Set up the form of the partial fraction decomposition. For the repeated linear factor \( x^2 \), include terms for each power: \( \frac{A}{x} + \frac{B}{x^2} \). For the irreducible quadratic factor \( x^2 + 4 \), include a linear numerator: \( \frac{Cx + D}{x^2 + 4} \). So the decomposition looks like: \[ \frac{x+4}{x^2 (x^2 + 4)} = \frac{A}{x} + \frac{B}{x^2} + \frac{Cx + D}{x^2 + 4} \]
3. Multiply both sides of the equation by the common denominator \( x^2 (x^2 + 4) \) to clear the fractions. This gives: \[ x + 4 = A x (x^2 + 4) + B (x^2 + 4) + (Cx + D) x^2 \]
4. Expand the right-hand side by distributing each term: \[ A x^3 + 4 A x + B x^2 + 4 B + C x^3 + D x^2 \]. Group like terms by powers of \( x \): \[ (A + C) x^3 + (B + D) x^2 + 4 A x + 4 B \]
5. Set up a system of equations by equating the coefficients of corresponding powers of \( x \) on both sides. On the left, the expression is \( x + 4 \), which can be written as \( 0 x^3 + 0 x^2 + 1 x + 4 \). So, equate coefficients: \[ A + C = 0, \quad B + D = 0, \quad 4 A = 1, \quad 4 B = 4 \]. Solve this system to find \( A, B, C, D \).

Key Concepts

Rational Expressions

A rational expression is a fraction where both the numerator and denominator are polynomials. Understanding how to manipulate these expressions is essential for simplifying, factoring, and decomposing them into partial fractions.

Partial Fraction Decomposition

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

Factoring Polynomials

Factoring polynomials means rewriting them as products of simpler polynomials. Recognizing factors like linear terms and irreducible quadratics is crucial for setting up the correct form of partial fractions.