/*! 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 18 Write the partial fraction decom... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 18

Write the partial fraction decomposition of each rational expression. $$\frac{4 x^{2}-5 x-15}{x(x+1)(x-5)}$$

Short Answer

Expert verified
The partial fraction decomposition of the given function is \(\frac{-5}{x} + \frac{3}{x+1} + \frac{-2}{x-5}\).

Step by step solution

01

Formulate the Decomposition

To write the partial fraction decomposition, we first need to identify the denominators of the fractions. In our rational expression, the denominator factors are \(x\), \(x+1\), and \(x-5\). So, we write our fraction as the sum of simpler fractions with these denominators. Let's represent the numerators of these simpler fractions (the coefficients we want to find) as \(A\), \(B\), and \(C\) respectively. So, the decomposition will look like: \[\frac{4x^2 - 5x - 15}{x(x+1)(x-5)} = \frac{A}{x} + \frac{B}{x+1} + \frac{C}{x-5}\]
02

Clear Fractions and Rearrange

This rearrangement will help us solve for coefficients \(A\), \(B\), and \(C\). Multiply the entire equation by the common denominator \(x(x+1)(x-5)\) to clear the fractions. This simplification results in: \[4x^2 - 5x - 15 = Ax(x+1)(x-5) + Bx(x)(x-5) + Cx(x)(x+1)\]
03

Distribute Terms and Group Like Terms

Fagine you are distributing each term on the right side. Once all terms are distributed, collect and group like terms. Our rearranged expression looks like:\[4x^2 - 5x - 15 = -5Ax + 5Bx - Ax^2 - Bx^2 + Ax + Bx - Cx^2 - 5Cx\]
04

Compare Coefficients and Set up Equations

In order to solve for \(A\), \(B\), and \(C\), we need to set up three separate equations by comparing the coefficients of like terms on both sides of the equation. Here are the equations we get from comparing coefficients: \[Equation 1: -A - B - C = 4\]\[Equation 2: -5A + 5B + A + B - 5C = -5\]\[Equation 3: A + B - C = -15\]
05

Solve the Simultaneous Equations

By solving these simultaneous equations, we find that the values of \(A\), \(B\), and \(C\) are -5, 3, and -2 respectively.
06

Write Out the Decomposed Fraction

Finally, we substitute the values for \(A\), \(B\), \(C\) back into our decomposition equation. The partial fraction decomposition of the given rational function is:\[\frac{4x^2 - 5x - 15}{x(x+1)(x-5)} = \frac{-5}{x} + \frac{3}{x+1} + \frac{-2}{x-5}\]

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 Exercises \(5-18\), solve each system by the substitution method. $$ \begin{aligned} &4 x+3 y=0\\\ &2 x-y=0 \end{aligned} $$

Use a system of linear equations to solve Exercises \(57-67\) The graph shows the calories in some favorite fast foods. Use the information in Exercises \(57-58\) to find the exact caloric content of the specified foods. (GRAPH CAN'T COPY) One pan pizza and two beef burritos provide 1980 calories. Two pan pizzas and one beef burrito provide 2670 calories. Find the caloric content of each item.

In Exercises \(5-18\), solve each system by the substitution method. $$ \begin{aligned} &x=3 y+7\\\ &x=2 y-1 \end{aligned} $$

How can you verify your result for the partial fraction decomposition for a given rational expression without using a graphing utility?

Find the partial fraction decomposition for \(\frac{2}{x(x+2)}\) and use the result to find the following sum: $$\frac{2}{1 \cdot 3}+\frac{2}{3 \cdot 5}+\frac{2}{5 \cdot 7}+\dots+\frac{2}{99 \cdot 101}$$
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation

Calculus

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.