/*! 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 57 Explain how to find the partial ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 57

Explain how to find the partial fraction decomposition of a rational expression with a repeated, prime quadratic factor in the denominator.

Short Answer

Expert verified
The method to find the partial fraction decomposition of a rational expression with a repeated, prime quadratic factor in the denominator is through a process starting with identifying the quadratic factor, setting up simpler fractions, matching terms with the original fraction, determining the coefficients and finally writing out the decomposition.

Step by step solution

01

Identify Repeated Quadratic Factor

Look at the denominator of the given rational expression. Identify the quadratic factor that is repeated in the denominator. Let's denote this factor as \(q(x)\).
02

Set-up Partial Fractions

In order to decompose the original fraction, set up a series of fractions: one for the first occurrence of the repeated quadratic factor \(q(x)\), and another for each subsequent occurrence up to the power n of the repeated factor. This will appear as \(A/q(x) + B/q(x)^2 + C/q(x)^3 + ... + Z/q(x)^n\). Note that each fraction's numerator is a variable that we will need to solve for.
03

Match Terms with Original Fraction

Now, equate the set-up fraction series to the original fraction. Multiply each side of the equation by the denominator of the original fraction to clear the denominators.
04

Determine Coefficients

From the equation obtained in step 3, you should have an equation in terms of x. Here, assign different suitable values to x to make the equation easier to solve. Next, solve the series of equations to find the values for A, B, C, ..., Z. These values are the coefficients for each term in the decomposition.
05

Write Out the Partial Fraction Decomposition

Once you have all the values for A, B, C, ..., Z, rewrite the decomposition, replacing the variables with their solved values. This is the partial fraction decomposition of the original rational expression.

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!

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

In your own words, describe how to solve a linear programming problem.

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 $_____.

Verify the identity: $$\frac{1}{\sin x \cos x}-\frac{\cos x}{\sin x}=\tan x$$

Solve the systems. $$ \left\\{\begin{array}{l} {\log x^{2}=y+3} \\ {\log x=y-1} \end{array}\right. $$

When a crew rows with the current, it travels 16 miles in 2 hours. Against the current, the crew rows 8 miles in 2 hours. Let \(x=\) the crew's rowing rate in still water and let \(y=\) the rate of the current. The following chart summarizes this information:
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Logic and Functions

Read Explanation

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

Read Explanation

Decision Maths

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.