/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 11 write the partial fraction decom... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 11

write the partial fraction decomposition of each rational expression. $$\frac{3 x+50}{(x-9)(x+2)}$$

Short Answer

Expert verified
The partial fraction decomposition of the rational expression \( \frac{3x+50}{(x-9)(x+2)} \) is \( -\frac{44}{x-9} + \frac{77}{x+2} \).

Step by step solution

01

Identify the Form of the Partial Fractions

Identify that the given rational function will break into two partial fractions of the form: \( \frac{A}{x-9} + \frac{B}{x+2} \) where A and B are constants to be found.
02

Equating to the Original Fraction

Equating \( \frac{3x+50}{(x-9)(x+2)} = \frac{A}{x-9} + \frac{B}{x+2} \), we can rearrange this to be \( 3x+50 = A(x+2) + B(x-9) \). This is now a system of linear equations that we'll be solving to find the values for A and B.
03

Solve for A

First, isolate for A by letting \( x=-2 \) since this will cause B to disappear. So the equation becomes \( 3(-2)+50 = A(-2+2) \), thus A equals to \(-44\).
04

Solve for B

Then, isolate for B by letting \( x=9 \) since this will cause A to disappear. This gives us the equation \( 3(9)+50 = B(9-9) \), thus B equals to \( 77 \).
05

Finalize the Decomposition

Substitute the found values of A and B to the previous form and the final partial fraction decomposition becomes: \( -\frac{44}{x-9} + \frac{77}{x+2} \).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rational Expressions
Rational expressions are fractions wherein both the numerator and the denominator are polynomials. In the exercise \( \frac{3x+50}{(x-9)(x+2)} \), the numerator is a linear polynomial \(3x+50\), and the denominator is the product of two linear factors \(x-9\) and \(x+2\).

To simplify complex rational expressions or to integrate them, partial fraction decomposition is utilized. It breaks down a complicated fraction into simpler 'partial' fractions, which makes further manipulation or integration much more manageable. When decomposing a function, we look for coefficients (labeled as A, B, etc.) that make the original expression equivalent to a sum of its simpler components. Identifying the correct form for partial fractions based on the factors in the denominator is essential.
Linear Equations
Linear equations are algebraic equations where each term is either a constant or the product of a constant and a single variable. A linear equation looks like this: \( ax + b = 0 \), where \( a \) and \( b \) are constants.

In our exercise, after setting up the equation in the form \( 3x+50 = A(x+2) + B(x-9) \), the goal is to find the constants A and B, which will satisfy the equation for all values of x. By choosing strategic values for x, you can create simpler, solvable linear equations. For instance, letting \( x=-2 \) gives a direct value for A since it eliminates the B term, and doing similarly for B by letting \( x=9 \). This step is essential as it simplifies the process of finding the constants needed for the partial fraction decomposition.
Systems of Equations
When we have more than one variable and more than one equation, we're dealing with a system of equations. In the context of partial fraction decomposition, after equating the original expression to the sum of fractions, we end up with an equation that must hold true for all x values. This gives us a system to solve.

From our example, \( 3x+50 = A(x+2) + B(x-9) \) offers a system with variables A and B. By strategically choosing values for x, we can solve this system 'by substitution,' which means we turn a system of equations into a single equation that we can solve since one variable gets eliminated. First, we find A by choosing \( x=-2 \), then we solve for B with \( x=9 \). This strategy streamlines the process of solving a system of equations that arises during partial fraction decomposition, making the task of finding the unknown constants more straightforward.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

will help you prepare for the material covered in the next section. Solve by the addition method: $$\left\\{\begin{array}{l}{2 x+4 y=-4} \\\\{3 x+5 y=-3}\end{array}\right.$$

a. A student earns \(\$ 15\) per hour for tutoring and \(\$ 10\) per hour as a teacher's aide. Let \(x=\) the number of hours each week spent tutoring and let \(y=\) the number of hours each week spent as a teacher's aide. Write the objective function that models total weekly earnings. b. The student is bound by the following constraints: \(\cdot\) To have enough time for studies, the student can work no more than 20 hours per week. \(\cdot\) The tutoring center requires that each tutor spend at least three hours per week tutoring. \(\cdot\) The tutoring center requires that each tutor spend no more than eight hours per week tutoring. Write a system of three inequalities that models these constraints. c. Graph the system of inequalities in part (b). Use only the first quadrant and its boundary, because \(x\) and \(y\) are nonnegative. d. Evaluate the objective function for total weekly earnings at each of the four vertices of the graphed region. [The vertices should occur at \((3,0),(8,0),(3,17),\) and \((8,12) .]\) e. Complete the missing portions of this statement: The student can earn the maximum amount per week by tutoring for ____ hours per week and working as a teacher's aide for ____ hours per week. The maximum amount that the student can earn each week is $_____.

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. Objective Function Constraints $$ \begin{aligned} &z=4 x+y\\\ &\left\\{\begin{array}{l} {x \geq 0, y \geq 0} \\ {2 x+3 y \leq 12} \\ {x+y \geq 3} \end{array}\right. \end{aligned} $$

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. Objective Function Constraints $$\begin{aligned} &z=3 x+2 y\\\ &\left\\{\begin{array}{c} {x \geq 0, y \geq 0} \\ {2 x+y \leq 8} \\ {x+y \geq 4} \end{array}\right. \end{aligned}$$

Members of the group should interview a business executive who is in charge of deciding the product mix for a business. How are production policy decisions made? Are other methods used in conjunction with linear programming? What are these methods? What sort of academic background, particularly in mathematics, does this executive have? Present a group report addressing these questions, emphasizing the role of linear programming for the business.
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation

Statistics

Read Explanation

Probability and Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.