Textbook Question
In Exercises 1–4, determine whether the given ordered pair is a solution of the system. (2, 5) 2x + 3y = 17 x + 4y = 16
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Ch. 5 - Systems of Equations and Inequalities
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Chapter 6, Problem 3
Write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants.
Verified step by step guidance1
Identify the denominator factors and their types. Here, the denominator is \((x+2)(x-3)^2\), which consists of a linear factor \((x+2)\) and a repeated linear factor \((x-3)^2\).
Write a separate term for each factor in the denominator. For the linear factor \((x+2)\), the corresponding term in the partial fraction decomposition is \(\frac{A}{x+2}\), where \(A\) is a constant to be determined.
For the repeated linear factor \((x-3)^2\), write terms for each power up to the multiplicity. This means you write \(\frac{B}{x-3} + \frac{C}{(x-3)^2}\), where \(B\) and \(C\) are constants to be determined.
Combine all terms to write the full form of the partial fraction decomposition as: \(\frac{A}{x+2} + \frac{B}{x-3} + \frac{C}{(x-3)^2}\).
Note that you do not need to solve for \(A\), \(B\), and \(C\); just write the form of the decomposition as shown.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Partial Fraction Decomposition
Partial fraction decomposition is a method used to express a rational function as a sum of simpler fractions whose denominators 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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Decomposition of Functions
Factorization of the Denominator
Understanding how to factor the denominator into linear and repeated factors is essential. In this problem, the denominator is already factored as (x + 2)(x - 3)^2, indicating a linear factor and a repeated linear factor, which affects the form of the partial fractions.
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02:58Rationalizing Denominators
Form of Partial Fractions for Repeated Factors
When the denominator contains repeated linear factors like (x - 3)^2, the partial fraction decomposition includes terms for each power of the repeated factor. Specifically, you write separate fractions with denominators (x - 3) and (x - 3)^2, each with its own constant numerator.
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04:36Factor by Grouping
Related Practice
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Solve by the method of your choice. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets.
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Textbook Question
Determine if the given ordered triple is a solution of the system.
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Find the value of the objective function at each corner of the graphed region. What is the maximum value of the objective function? What is the minimum value of the objective function? 1. Objective Function z=40x+50y
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In Exercises 1–18, solve each system by the substitution method.
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Textbook Question
Graph each inequality. x−2y>10
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