In Exercises 9–42, write the partial fraction decomposition of each rational expression. x^3+x^2+2/(x² + 2)²

Verified step by step guidance

1. Step 1: Identify the form of the partial fraction decomposition.
2. Step 2: Set up the partial fraction decomposition.
3. Step 3: Multiply through by the common denominator to clear the fractions.
4. Step 4: Expand and collect like terms on both sides of the equation.
5. Step 5: Equate coefficients for corresponding powers of x to form a system of equations.

Key Concepts

Rational Expressions

A rational expression is a fraction where both the numerator and the denominator are polynomials. Understanding rational expressions is crucial for performing operations like addition, subtraction, and decomposition. In this case, the expression involves a polynomial in the numerator and a polynomial raised to a power in the denominator.

Partial Fraction Decomposition

Partial fraction decomposition is a method used to express a rational function as a sum of simpler fractions. This technique is particularly useful for integrating rational functions or simplifying complex expressions. The goal is to break down the original expression into fractions whose denominators are factors of the original denominator.

Polynomial Factorization

Polynomial factorization involves breaking down a polynomial into its constituent factors, which can be linear or irreducible quadratic expressions. This is essential for partial fraction decomposition, as it allows us to identify the appropriate form of the simpler fractions. In the given expression, recognizing the structure of the denominator is key to determining the correct decomposition.