Ch. 5 - Systems of Equations and Inequalities

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 5 - Systems of Equations and Inequalities

Problem 4

Chapter 6, Problem 4

In Exercises 1–8, write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants. (3x+16)/(x + 1) (x − 2)²

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Identify the factors in the denominator: (x + 1) and (x - 2)². Note that (x - 2) is a repeated linear factor with multiplicity 2.

Write a separate term for each factor and each power of the repeated factor. For the linear factor (x + 1), use a term of the form

A / (x + 1)

where A is a constant to be determined.

For the repeated linear factor (x - 2)², write terms for each power: one term with denominator (x - 2) and another with denominator (x - 2)². These take the form

B / (x - 2)

and

C / (x - 2)^{2}

respectively, where B and C are constants.

Combine all terms to write the full partial fraction decomposition:

\frac{A}{x} + \frac{B}{x} + \frac{C}{(^{x - 2) 2}}

This expression represents the form of the partial fraction decomposition. Constants A, B, and C would be found by multiplying both sides by the denominator and equating coefficients, but that is not required here.

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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, making integration or other operations easier. It involves breaking down a complex fraction into a sum of fractions with simpler denominators, typically linear or quadratic factors.

Factorization of Denominators

Understanding how to factor the denominator into linear and repeated factors is essential. In this problem, the denominator is factored as (x + 1)(x − 2)², indicating one linear factor and one repeated linear factor, which affects the form of the partial fractions.

Form of Partial Fractions for Repeated Factors

When a denominator has repeated linear factors like (x − 2)², the partial fraction decomposition includes terms for each power of the repeated factor. For (x − 2)², the decomposition includes terms with denominators (x − 2) and (x − 2)², each with its own constant numerator.

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Related Practice

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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Textbook Question

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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Textbook Question

An objective function and a system of linear inequalities representing constraints are given. a. Graph the system of inequalities representing the constraints. b. Find the value of the objective function at each corner of the graphed region. c. Use the values in part (b) to determine the maximum value of the objective function and the values of x and y for which the maximum occurs.

Textbook Question

Write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants. $\frac{5x^2 - 6x + 7}{(x - 1)(x^2 + 1)}$

Textbook Question

Graph each inequality. x−2y>10

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Textbook Question

In Exercises 1–18, solve each system by the substitution method. $\begin{cases}\)y = x^2 - 4x - 10 $\y$ = -x^2 - 2x + 14\(\end{cases}$