In Exercises 1–8, write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants. x^3 + x² /(x² + 4)^2

Verified step by step guidance

1. Step 1: Identify the form of the rational expression.
2. Step 2: Recognize that the denominator is a repeated irreducible quadratic factor.
3. Step 3: Set up the partial fraction decomposition.
4. Step 4: Assign constants to each term in the decomposition.
5. Step 5: Write the partial fraction decomposition expression.

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 such as addition, subtraction, multiplication, and division, as well as for decomposing them into simpler components, which is the focus of this question.

Partial Fraction Decomposition

Partial fraction decomposition is a technique used to express a rational function as a sum of simpler fractions. This method is particularly useful for integrating rational functions or simplifying complex expressions. The process involves breaking down the rational expression based on the factors of the denominator, which can include linear and irreducible quadratic factors.

Polynomial Degree and Factorization

The degree of a polynomial is the highest power of the variable in the expression. In the context of partial fraction decomposition, understanding the degree helps in determining the form of the decomposition. Additionally, factorization of the denominator into linear and quadratic factors is essential, as it dictates the structure of the partial fractions that will be formed.