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Ch. 5 - Systems of Equations and Inequalities
Blitzer - College Algebra 8th Edition
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Tutti i libri di testoBlitzer 8th EditionCh. 5 - Systems of Equations and InequalitiesProblema 29
Capitolo 6, Problema 29

Write the partial fraction decomposition of each rational expression. 5x2 -6x+7/(x − 1) (x2 + 1)

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Identify the form of the denominator. Here, the denominator is \( (x - 1)(x^2 + 1) \), which consists of a linear factor \( (x - 1) \) and an irreducible quadratic factor \( (x^2 + 1) \).
Set up the partial fraction decomposition with unknown constants. For the linear factor, use a constant numerator \( A \), and for the quadratic factor, use a linear numerator \( Bx + C \). So, write: \[ \frac{5x^2 - 6x + 7}{(x - 1)(x^2 + 1)} = \frac{A}{x - 1} + \frac{Bx + C}{x^2 + 1} \]
Multiply both sides of the equation by the denominator \( (x - 1)(x^2 + 1) \) to clear the fractions: \[ 5x^2 - 6x + 7 = A(x^2 + 1) + (Bx + C)(x - 1) \]
Expand the right-hand side by distributing \( A \) and then expanding \( (Bx + C)(x - 1) \). This will give a polynomial expression in terms of \( x \): \[ A x^2 + A + Bx^2 - Bx + Cx - C \]
Group like terms (powers of \( x \)) on the right side and equate the coefficients of corresponding powers of \( x \) from both sides to form a system of equations. Solve this system to find the values of \( A \), \( B \), and \( C \).

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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 expressions into manageable parts.
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Decomposition of Functions

Factoring the Denominator

Factoring the denominator involves expressing it as a product of linear and/or irreducible quadratic factors. In this problem, the denominator is already factored as (x − 1)(x² + 1), which guides the form of the partial fractions to include terms over each factor.
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Rationalizing Denominators

Form of Partial Fractions for Linear and Quadratic Factors

For linear factors like (x − 1), the partial fraction takes the form A/(x − 1). For irreducible quadratic factors like (x² + 1), the partial fraction is expressed as (Bx + C)/(x² + 1). This ensures the decomposition accounts for all possible numerators matching the degree of the factors.
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Solving Linear Equations with Fractions
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