Download the App!

Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.

Sent to:
Search glass icon
  • Login
  • Textbooks
  • Ask our Educators
  • Study Tools
    Study Groups Bootcamps Quizzes AI Tutor iOS Student App Android Student App StudyParty
  • For Educators
    Become an educator Educator app for iPad Our educators
  • For Schools

Problem

Write out the form of the partial fraction decomp…

06:37

Question

Answered step-by-step

Problem 4 Medium Difficulty

Write out the form of the partial fraction decomposition of the function (as in Example 7). Do not determine the numerical values of the coefficients.

(a) $ \dfrac{x^4 - 2x^3 + x^2 + 2x - 1}{x^2 - 2x + 1} $ (b) $ \dfrac{x^2 - 1}{x^3 + x^2 + x} $


Video Answer

Solved by verified expert

preview
Numerade Logo

This problem has been solved!

Try Numerade free for 7 days

JH
J Hardin
Numerade Educator

Like

Report

Textbook Answer

Official textbook answer

Video by J Hardin

Numerade Educator

This textbook answer is only visible when subscribed! Please subscribe to view the answer

Related Courses

Calculus 2 / BC

Calculus: Early Transcendentals

Chapter 7

Techniques of Integration

Section 4

Integration of Rational Functions by Partial Fractions

Related Topics

Integration Techniques

Discussion

You must be signed in to discuss.
Top Calculus 2 / BC Educators
Heather Zimmers

Oregon State University

Kristen Karbon

University of Michigan - Ann Arbor

Samuel Hannah

University of Nottingham

Michael Jacobsen

Idaho State University

Calculus 2 / BC Courses

Lectures

Video Thumbnail

01:53

Integration Techniques - Intro

In mathematics, integration is one of the two main operations in calculus, with its inverse, differentiation, being the other. Given a function of a real variable, an antiderivative, integral, or integrand is the function's derivative, with respect to the variable of interest. The integrals of a function are the components of its antiderivative. The definite integral of a function from a to b is the area of the region in the xy-plane that lies between the graph of the function and the x-axis, above the x-axis, or below the x-axis. The indefinite integral of a function is an antiderivative of the function, and can be used to find the original function when given the derivative. The definite integral of a function is a single-valued function on a given interval. It can be computed by evaluating the definite integral of a function at every x in the domain of the function, then adding the results together.

Video Thumbnail

27:53

Basic Techniques

In mathematics, a technique is a method or formula for solving a problem. Techniques are often used in mathematics, physics, economics, and computer science.

Join Course
Recommended Videos

06:16

Write out the form of the …

01:54

Write out the form of the …

02:40

Write out the form of the …

06:37

Write out the form of the …

03:17

Write out the form of the …

02:34

Write out the form of the …

01:18

Write out the form of the …

01:42

Write out the form of the …

02:15

Write out the form of the …

09:45

Write out the form of the …

02:21

Write out the form of the …

04:26

Write out the form of the …

04:21

Write out the form of the …

00:30

Write down the form of the…

00:59

Write out the form of the …

Watch More Solved Questions in Chapter 7

Problem 1
Problem 2
Problem 3
Problem 4
Problem 5
Problem 6
Problem 7
Problem 8
Problem 9
Problem 10
Problem 11
Problem 12
Problem 13
Problem 14
Problem 15
Problem 16
Problem 17
Problem 18
Problem 19
Problem 20
Problem 21
Problem 22
Problem 23
Problem 24
Problem 25
Problem 26
Problem 27
Problem 28
Problem 29
Problem 30
Problem 31
Problem 32
Problem 33
Problem 34
Problem 35
Problem 36
Problem 37
Problem 38
Problem 39
Problem 40
Problem 41
Problem 42
Problem 43
Problem 44
Problem 45
Problem 46
Problem 47
Problem 48
Problem 49
Problem 50
Problem 51
Problem 52
Problem 53
Problem 54
Problem 55
Problem 56
Problem 57
Problem 58
Problem 59
Problem 60
Problem 61
Problem 62
Problem 63
Problem 64
Problem 65
Problem 66
Problem 67
Problem 68
Problem 69
Problem 70
Problem 71
Problem 72
Problem 73
Problem 74
Problem 75

Video Transcript

let's find the partial fraction decomposition for both of these functions. And there's no need to determine the numerical values of the coefficients. So looking at the first fraction, the first thing that I noticed is that the numerator is a higher degree, then the denominator. So any time the numerator has degree equal to or larger than the denominator, the first thing we should always do is long division, our polynomial division. And that's what we have set up over here on the right. So we'd liketo find the quotient here. So we have an X square exit, the forest. So we're gonna x squared up here, and then we multiply two x cubed plus X square, and then we subtract the whole thing doing so we have a lot of cancellation here, okay? And then we're just left with two X minus one. Now, this quadratic that we're dividing by does not go into to explain this one. So we have for part A. We can rewrite this as the question which is X square, and then we have the remainder over. The thing that we originally divided by this is why the long division is useful now every time we finished long division that they nominator will have larger degree than the numerator. And now we do partial fraction the composition. So as before, the first thing always look at that denominator. See if it can be factored here we can factor. This is X minus one squared. So this is what the book calls case, too. This's the repeated linear factor, which in this case, is just X minus one. So using the formula for case too x square, this's not a fraction less his write that as it is and then for this fraction we have a over X minus one plus be over. It's not this one square, and that will be our partial fraction. The composition for party. Let's go to the next page for party. So for me X squared minus one X cube X square in the next. So we noticed that the denominator has degree larger than the numerator. That's good. Not division necessary. Might be helpful to factor the numerator, but really more concerned with the denominator unless we could factor things out. So maybe I should go ahead and actually do this. Not always useful, but it can't be helpful. Sometimes you might be able to cancel things, though, but in this case, it doesn't look like that will happen in the denominator. Let's take out a X and there's our factories ation. This term, we can't factor. It's a linear. But then the second turn this may factor into two Lanier's So we need to look at the discriminative B squared minus four a. C Yeah, or again, thes coefficients are coming from just some polynomial this form. All right, In our problem, we see that a equals B equals C equals one finger. So that B squared minus four A. C is negative three, which is less than zero. And that tells us that this polynomial does not factor this guy appear. So that puts us in Case three. This is the case in which we just have a quadratic that doesn't factor not repeated. So this is what the text calling the textbook is calling Case three. No, For the first term, I see a X. So I just have a constant over X and then for the next term, because it's a quadratic on the bottom, it's not enough to have just a constant up top. You need a linear polynomial. So that's where I'Ll put the explosive and there's our answer for party.

Get More Help with this Textbook
James Stewart

Calculus: Early Transcendentals

View More Answers From This Book

Find Another Textbook

Study Groups
Study with other students and unlock Numerade solutions for free.
Math (Geometry, Algebra I and II) with Nancy
Arrow icon
Participants icon
142
Hosted by: Ay?Enur Çal???R
Math (Algebra 2 & AP Calculus AB) with Yovanny
Arrow icon
Participants icon
68
Hosted by: Alonso M
See More

Related Topics

Integration Techniques

Top Calculus 2 / BC Educators
Heather Zimmers

Oregon State University

Kristen Karbon

University of Michigan - Ann Arbor

Samuel Hannah

University of Nottingham

Michael Jacobsen

Idaho State University

Calculus 2 / BC Courses

Lectures

Video Thumbnail

01:53

Integration Techniques - Intro

In mathematics, integration is one of the two main operations in calculus, with its inverse, differentiation, being the other. Given a function of a real variable, an antiderivative, integral, or integrand is the function's derivative, with respect to the variable of interest. The integrals of a function are the components of its antiderivative. The definite integral of a function from a to b is the area of the region in the xy-plane that lies between the graph of the function and the x-axis, above the x-axis, or below the x-axis. The indefinite integral of a function is an antiderivative of the function, and can be used to find the original function when given the derivative. The definite integral of a function is a single-valued function on a given interval. It can be computed by evaluating the definite integral of a function at every x in the domain of the function, then adding the results together.

Video Thumbnail

27:53

Basic Techniques

In mathematics, a technique is a method or formula for solving a problem. Techniques are often used in mathematics, physics, economics, and computer science.

Join Course
Recommended Videos

06:16

Write out the form of the partial fraction decomposition of the function (as in…

01:54

Write out the form of the partial fraction decomposition of the function (as in…

02:40

Write out the form of the partial fraction decomposition of the function (as in…

06:37

Write out the form of the partial fraction decomposition of the function (as in…

03:17

Write out the form of the partial fraction decomposition of the function (as in…

02:34

Write out the form of the partial fraction decomposition of the function (as in…

01:18

Write out the form of the partial fraction decomposition of the function (as in…

01:42

Write out the form of the partial fraction decomposition of the function (as in…

02:15

Write out the form of the partial fraction decomposition of the function (as in…

09:45

Write out the form of the partial fraction decomposition of the function (as in…

02:21

Write out the form of the partial fraction decomposition of the function (as in…

04:26

Write out the form of the partial fraction decomposition of the function (as in…

04:21

Write out the form of the partial fraction decomposition of the function (as in…

00:30

Write down the form of the partial fraction decomposition of the given rational…

00:59

Write out the form of the partial fraction expansion of the function. Do not de…

Add To Playlist

Hmmm, doesn't seem like you have any playlists. Please add your first playlist.

Create a New Playlist

`

Share Question

Copy Link

OR

Enter Friends' Emails

Report Question

Get 24/7 study help with our app

 

Available on iOS and Android

About
  • Our Story
  • Careers
  • Our Educators
  • Numerade Blog
Browse
  • Bootcamps
  • Books
  • Notes & Exams NEW
  • Topics
  • Test Prep
  • Ask Directory
  • Online Tutors
  • Tutors Near Me
Support
  • Help
  • Privacy Policy
  • Terms of Service
Get started