In Exercises 16–24, write the partial fraction decomposition of each rational expression. (4x^2 - 3x - 4)/x(x + 2)(x - 1)

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1. Step 1: Recognize that the given rational expression \((4x^2 - 3x - 4) / (x(x + 2)(x - 1))\) is proper, meaning the degree of the numerator is less than the degree of the denominator. This allows us to proceed with partial fraction decomposition.
2. Step 2: Set up the partial fraction decomposition. Since the denominator \(x(x + 2)(x - 1)\) consists of three distinct linear factors, the decomposition will take the form: \(\frac{A}{x} + \frac{B}{x + 2} + \frac{C}{x - 1}\), where \(A\), \(B\), and \(C\) are constants to be determined.
3. Step 3: Combine the fractions on the right-hand side over a common denominator \(x(x + 2)(x - 1)\). This gives: \(\frac{A(x + 2)(x - 1) + Bx(x - 1) + Cx(x + 2)}{x(x + 2)(x - 1)}\).
4. Step 4: Equate the numerators of the original fraction and the combined fraction. This results in the equation: \(4x^2 - 3x - 4 = A(x + 2)(x - 1) + Bx(x - 1) + Cx(x + 2)\).
5. Step 5: Expand the terms on the right-hand side and collect like terms for \(x^2\), \(x\), and the constant. Then, equate the coefficients of \(x^2\), \(x\), and the constant on both sides of the equation to form a system of linear equations. 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 (4x^2 - 3x - 4)/x(x + 2)(x - 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 expressions or simplifying complex algebraic fractions. 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.

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of its factors. This is essential for partial fraction decomposition, as the decomposition relies on the factors of the denominator. In the given expression, x(x + 2)(x - 1) must be factored correctly to identify the appropriate form for the partial fractions, which typically includes constants or linear terms corresponding to each factor.