/*! 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 7 Write the partial fraction decom... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 7

Write the partial fraction decomposition for the expression. $$ \frac{3 x^{2}-2 x-5}{x^{3}+x^{2}} $$

Short Answer

Expert verified
\(\frac{3x-5}{x^{2}}\)

Step by step solution

01

Factor the Denominator

The denominator of the provided expression can be factored using the common factor rule. This rule states that if two terms share a common factor, this factor can be taken out from these terms. The denominator \(x^{3}+x^{2}\) can be factored to be \(x^{2}(x+1)\).
02

Decompose into Partial Fractions

A partial fraction decomposition of the fraction is done by expressing the original fraction as a sum of simpler fractions. The numerator of the original fraction dictates the numerator of the simpler fractions. Given that the denominator of the included expression has been factored into \(x^{2}(x+1)\), the expression would be decomposed into \(\frac{Ax+B}{x^{2}} + \frac{C}{x+1}\) where A, B and C are constants which we will calculate in the next steps.
03

Equate coefficients and Solve for A, B and C

The right hand side should be identical to the original function. Consequently, multiplying out the right hand side will give \((Ax+B)(x+1) + Cx^{2}\) which simplifies to \(Ax^{2} + Bx + Ax + B + Cx^{2}\), then \(Ax^{2} + Cx^{2} + (A+B)x + B\), and finally \((A+C)x^{2} + (A+B)x + B\). On the left hand side, the coefficients of the polynomial are \(3x^{2} - 2x - 5\). By equating the coefficients on both sides, we can find the values for A, B and C. These equations will be: 1. A + C = 3, 2. A + B = -2, 3. B = -5. From 3., we deduce that B = -5. By substituting B into equation 2., we find A = -2 - B = -2 - (-5) = 3. Finally, by substituting A into equation 1., we find C = 3 - A = 3 - 3 = 0
04

Substitute A, B and C into the partial fractions

Substitute the values A=3, B=-5, and C=0 into the partial fractions \(\frac{Ax+B}{x^{2}} + \frac{C}{x+1}\) to give us \(\frac{3x-5}{x^{2}} + \frac{0}{x+1}\). Since \(\frac{0}{x+1}\) is essentially 0, this term will drop out, leaving our final answer as \(\frac{3x-5}{x^{2}}\).

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!

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

Population Growth \(\ln\) Exercises 57 and 58, use a graphing utility to graph the growth function. Use the table of integrals to find the average value of the growth function over the interval, where \(N\) is the size of a population and \(t\) is the time in days. $$ N=\frac{375}{1+e^{4.20-0.25 t}}, \quad[21,28] $$

Present Value In Exercises 25 and 26, use a program similar to the Simpson's Rule program on page 906 with \(n=8\) to approximate the present value of the income \(c(t)\) over \(t_{1}\) years at the given annual interest rate \(r\). Then use the integration capabilities of a graphing utility to approximate the present value. Compare the results. (Present value is defined in Section 12.1.) $$ c(t)=200,000+15,000 \sqrt[3]{t}, r=10 \%, t_{1}=8 $$

Determine whether the improper integral diverges or converges. Evaluate the integral if it converges. $$ \int_{-\infty}^{0} e^{-x} d x $$

Consumer and Producer Surpluses Find the consumer surplus and the producer surplus for a product with the given demand and supply functions. Demand: \(p=\frac{60}{\sqrt{x^{2}+81}}\), Supply: \(p=\frac{x}{3}\)

Present Value In Exercises 25 and 26, use a program similar to the Simpson's Rule program on page 906 with \(n=8\) to approximate the present value of the income \(c(t)\) over \(t_{1}\) years at the given annual interest rate \(r\). Then use the integration capabilities of a graphing utility to approximate the present value. Compare the results. (Present value is defined in Section 12.1.) $$ c(t)=6000+200 \sqrt{t}, r=7 \%, t_{1}=4 $$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Calculus

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.