In Exercises 16–24, write the partial fraction decomposition of each rational expression.3x/(x - 2)(x^2 + 1)

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.
5. Step 5: Equate coefficients to solve for unknowns.

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 3x/((x - 2)(x^2 + 1)) is a rational expression that needs to be decomposed 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 in the denominator determines how the rational expression can be decomposed. In the given expression, recognizing (x - 2) as a linear factor and (x^2 + 1) as an irreducible quadratic is key to the decomposition process.