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Ch. 5 - Systems of Equations and Inequalities
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 5 - Systems of Equations and InequalitiesProblem 17
Chapter 6, Problem 17

Write the partial fraction decomposition of each rational expression. 4x2+13x-9/x (x − 1)(x+3)

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Identify the form of the rational expression. Since the denominator is factored as \(x(x - 1)(x + 3)\), and all factors are linear and distinct, the partial fraction decomposition will have terms of the form \(\frac{A}{x} + \frac{B}{x - 1} + \frac{C}{x + 3}\).
Set up the equation by expressing the given rational expression as a sum of partial fractions: \(\frac{4x^{2} + 13x - 9}{x(x - 1)(x + 3)} = \frac{A}{x} + \frac{B}{x - 1} + \frac{C}{x + 3}\).
Multiply both sides of the equation by the common denominator \(x(x - 1)(x + 3)\) to clear the denominators: \(4x^{2} + 13x - 9 = A(x - 1)(x + 3) + B x (x + 3) + C x (x - 1)\).
Expand the right-hand side by multiplying out each term: - \(A(x - 1)(x + 3) = A(x^{2} + 2x - 3)\) - \(B x (x + 3) = B(x^{2} + 3x)\) - \(C x (x - 1) = C(x^{2} - x)\) Then combine all terms to get a polynomial in standard form.
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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Rational Expressions

A rational expression is a fraction where both the numerator and denominator are polynomials. Understanding how to manipulate these expressions, including factoring and simplifying, is essential before performing partial fraction decomposition.
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Partial Fraction Decomposition

Partial fraction decomposition breaks a complex rational expression into a sum of simpler fractions with simpler denominators. This technique is useful for integration and solving equations involving rational expressions.
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Factoring Polynomials

Factoring involves expressing a polynomial as a product of its factors. For partial fraction decomposition, the denominator must be factored into linear or irreducible quadratic factors to set up the correct form of the decomposition.
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