Q. 35 In Problem, Write the partial fr... [FREE SOLUTION]

91影视

Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 11: Q. 35 (page 794)

In Problem, Write the partial fraction decomposition of each rational expression.
$\frac{x - 4}{x^{2} (x - 1)}$

Short Answer

Expert verified

Partial fraction decomposition of a rational expression $\frac{x - 4}{x^{2} (x - 1)}$ is

$\frac{x - 4}{x^{2} (x - 1)} = \frac{3}{x} + \frac{4}{x^{2}} + \frac{- 3}{(x - 1)}$

Step by step solution

Step 1. Given data

The given rational expression is

$\frac{x - 4}{x^{2} (x - 1)}$

Step 2. Decomposition of rational expression

Partial fraction decomposition of rational expression

$\frac{P (x)}{Q (x)} = \frac{A_{1}}{x - a_{1}} + \frac{A_{2}}{x - a_{2}} + 鈰� + \frac{A_{n}}{x - a_{n}} \frac{x - 4}{x^{2} (x - 1)} = \frac{A}{x} + \frac{B}{x^{2}} + \frac{C}{(x - 1)} \frac{x - 4}{x^{2} (x - 1)} = \frac{A x (x - 1)}{x^{2} (x - 1)} + \frac{B (x - 1)}{x^{2} (x - 1)} + \frac{C x^{2}}{x^{2} (x - 1)}$

Step 3. Value of numerator of partial fractions

Rearrange the equation

$\frac{x - 4}{x^{2} (x - 1)} = \frac{A x (x - 1)}{x^{2} (x - 1)} + \frac{B (x - 1)}{x^{2} (x - 1)} + \frac{C x^{2}}{x^{2} (x - 1)} x - 4 = A x (x - 1) + B (x - 1) + C x^{2} 0 x^{2} + 1 x - 4 = (A + C) x^{2} + (B - A) x - B$

Equate the constants

$B = 4$

Equate the coefficients ofx

$B - A = 1 4 - A = 1 A = 3$

Equate the coefficients of $x^{2}$

$A + C = 0 3 + C = 0 C = - 3$

So partial fraction decomposition isrole="math" localid="1646939664226" $\frac{x - 4}{x^{2} (x - 1)} = \frac{3}{x} + \frac{4}{x^{2}} + \frac{- 3}{(x - 1)}$

Step 4. Verification

Simplify the partial fractions

$\frac{3}{x} + \frac{4}{x^{2}} + \frac{- 3}{(x - 1)} = \frac{3 x (x - 1) + 4 (x - 1) - 3 x^{2}}{x^{2} (x - 1)} = \frac{3 x^{2} - 3 x + 4 x - 4 - 3 x^{2}}{x^{2} (x - 1)} = \frac{x - 4}{x^{2} (x - 1)}$

It is equal to the original expression so partial fraction decomposition is correct.

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

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

In Problem, Write the partial fraction decomposition of each rational expression.
$\frac{x - 4}{x^{2} (x - 1)}$

Short Answer

Step by step solution

Step 1. Given data

Step 2. Decomposition of rational expression

Step 3. Value of numerator of partial fractions

Step 4. Verification

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Decision Maths

Theoretical and Mathematical Physics

Applied Mathematics

Pure Maths

Probability and Statistics

Study anywhere. Anytime. Across all devices.

91影视

In Problem, Write the partial fraction decomposition of each rational expression.x-4x2(x-1)

Short Answer

Step by step solution

Step 1. Given data

Step 2. Decomposition of rational expression

Step 3. Value of numerator of partial fractions

Step 4. Verification

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Decision Maths

Theoretical and Mathematical Physics

Applied Mathematics

Pure Maths

Probability and Statistics

Study anywhere. Anytime. Across all devices.

In Problem, Write the partial fraction decomposition of each rational expression.
$\frac{x - 4}{x^{2} (x - 1)}$