/*! 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 46 Explain how to find the partial ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 46

Explain how to find the partial fraction decomposition of a rational expression with distinct linear factors in the denominator.

Short Answer

Expert verified
The main idea behind this problem is the concept of 'partial fractions', where a complex fraction is expressed as a sum of simpler fractions. The general steps involve factorizing the denominator, setting up partial fractions, combining like terms, equating and solving to find constants, and finally, representing the original fraction as a sum of simpler partial fractions.

Step by step solution

01

Factorization of the Denominator

Start by factorizing the denominator of the rational function. For instance, the denominator \(x^2 - 3x + 2\) can be factored as \((x - 1)(x - 2)\).
02

Setup Partial Fractions

Create a setup for partial fractions. Based on the factors found in Step 1, set up the partial fractions. For example, for our problem \(\frac{8x + 7}{x^2 - 3x + 2}\) will be set up as \(\frac{A}{x - 1} + \frac{B}{x - 2}\), where A and B are constants that we need to discover.
03

Combine Fractions

The next step is to combine these fractions. That would give \(\frac{A(x - 2) + B(x - 1)}{x^2 - 3x + 2}\) which should equal \(\frac{8x + 7}{x^2 - 3x + 2}\)
04

Solve for Constants

As these two rational fractions are equal, their numerators should be equal. That is, \(A(x - 2) + B(x - 1) = 8x + 7\). Now solve for A and B by plugging in appropriate x values (the roots of the denominator). This will let you solve for these constants.
05

Insert Constants into Set-up

Finally, we put these constants into the initial setup from step 2, and obtain the desired partial fractions decomposition.

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

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

In Exercises \(31-42,\) 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. \(x=9-2 y\) \(x+2 y=13\)

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

Explain how to graph \(2 x-3 y<6\).

Describe how to solve a system of inequalities.
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Logic and Functions

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.