Find the partial fraction decomposition for each rational expression. See Examples 1–4. (-3)/(x^2(x^2 + 5))

Verified step by step guidance

1. Step 1: Identify the form of the rational expression. In this case, the expression is in the form of a proper rational function because the degree of the numerator is less than the degree of the denominator.
2. Step 2: Identify the factors of the denominator. The denominator has two factors: x^2 and (x^2 + 5).
3. Step 3: Write the expression as a sum of partial fractions. Each fraction will have a denominator of one of the factors and a numerator of a constant or a polynomial of degree less than the denominator. In this case, the expression can be written as A/x + B/x^2 + C/(x^2 + 5), where A, B, and C are constants to be determined.
4. Step 4: Multiply both sides of the equation by the common denominator to clear the fractions. This will result in an equation that can be solved for A, B, and C.
5. Step 5: Solve the resulting equation for A, B, and C. This can be done by equating coefficients for like terms on both sides of the equation and solving the resulting system of linear 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 context, the expression (-3)/(x^2(x^2 + 5)) is a rational expression that requires decomposition into simpler fractions.

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 given rational expression into components that are easier to work with, based on the factors of the 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 the form of the factors determines how the decomposition is structured. In the given expression, x^2 and (x^2 + 5) are the factors that will guide the decomposition process.