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Ch. 5 - Systems of Equations and Inequalities
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Blitzer 8th EditionCh. 5 - Systems of Equations and Inequalities문제 18
6장, 문제 18

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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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.
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.
Step 3: Combine the fractions on the right-hand side over a common denominator \(x(x + 2)(x - 1)\). This gives: \(\frac{A(x + 2)(x - 1) + Bx(x - 1) + Cx(x + 2)}{x(x + 2)(x - 1)}\).
Step 4: Equate the numerators of the original fraction and the combined fraction. This results in the equation: \(4x^2 - 3x - 4 = A(x + 2)(x - 1) + Bx(x - 1) + Cx(x + 2)\).
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 \(C\).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.
추천 영상:
02:58
Rationalizing Denominators

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.
추천 영상:
4:07
Decomposition of Functions

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.
추천 영상:
07:30
Introduction to Factoring Polynomials
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