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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 19
6장, 문제 19

Write the partial fraction decomposition of each rational expression. 4x2 - 7x - 3/(x3 -x)

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First, factor the denominator \(x^3 - x\). Notice that you can factor out an \(x\) to get \(x(x^2 - 1)\).
Recognize that \(x^2 - 1\) is a difference of squares, so factor it further as \(x(x - 1)(x + 1)\).
Set up the partial fraction decomposition with unknown constants for each factor in the denominator: \(\frac{4x^2 - 7x - 3}{x(x - 1)(x + 1)} = \frac{A}{x} + \frac{B}{x - 1} + \frac{C}{x + 1}\).
Multiply both sides of the equation by the common denominator \(x(x - 1)(x + 1)\) to clear the fractions, resulting in: \(4x^2 - 7x - 3 = A(x - 1)(x + 1) + B x (x + 1) + C x (x - 1)\).
Expand the right side and collect like terms to form a polynomial equation. Then, equate the coefficients of corresponding powers of \(x\) on both sides to create a system of equations to solve for \(A\), \(B\), and \(C\).

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주요 개념

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

Partial Fraction Decomposition

Partial fraction decomposition is a method used to express a rational function as a sum of simpler fractions with denominators that are factors of the original denominator. This technique simplifies integration and other algebraic operations by breaking down complex fractions into manageable parts.
추천 영상:
4:07
Decomposition of Functions

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of its factors. For partial fractions, factoring the denominator completely into linear and/or irreducible quadratic factors is essential to set up the correct form of the decomposition.
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Introduction to Factoring Polynomials

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After expressing the rational function as a sum of partial fractions, you equate the original numerator to the combined numerator of the decomposed fractions. This leads to a system of equations for the unknown coefficients, which you solve to find the values that complete the decomposition.
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