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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 19
Capitolo 6, Problema 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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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.
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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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Setting Up and Solving Equations for Coefficients

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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