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

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1. Step 1: Recognize that the given rational expression is 3x / ((x - 2)(x^2 + 1)). The denominator is already factored into two distinct terms: a linear factor (x - 2) and an irreducible quadratic factor (x^2 + 1).
2. Step 2: Set up the partial fraction decomposition. For the linear factor (x - 2), assign a constant numerator A. For the irreducible quadratic factor (x^2 + 1), assign a linear numerator Bx + C. The decomposition will look like: 3x / ((x - 2)(x^2 + 1)) = A / (x - 2) + (Bx + C) / (x^2 + 1).
3. Step 3: Combine the right-hand side into a single fraction. To do this, find a common denominator, which is (x - 2)(x^2 + 1). Rewrite the terms as: A(x^2 + 1) + (Bx + C)(x - 2) / ((x - 2)(x^2 + 1)).
4. Step 4: Expand the numerator. Distribute A across (x^2 + 1) and distribute (Bx + C) across (x - 2). This gives: A(x^2) + A + Bx(x) - Bx(2) + C(x) - C(2). Combine like terms in the numerator.
5. Step 5: Equate the numerator of the combined fraction to the numerator of the original fraction, 3x. This will give you a system of equations to solve for A, B, and C by matching coefficients of like terms (x^2, x, and the constant term). Solve this system to find the values of A, B, and C.

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. Recognizing the types of factors helps in setting up the correct form for the partial fractions.