/*! 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 41 Write the partial fraction decom... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 41

Write the partial fraction decomposition of each rational expression. $$\frac{4 x^{2}+3 x+14}{x^{3}-8}$$

Short Answer

Expert verified
After solving the equation for A, B, and C, their calculated values represent the constants which will be substituted back into the equation, \[A/x + (Bx+C)/((x^2 - 2x + 4)\] to obtain the partial fraction decomposition. This concludes the exercise.

Step by step solution

01

Identify the format of the partial fractions

The denominator can be factored into \(x(x^2 - 2x + 4)\). Following this, the partial fraction decomposition of the expression can be written in the form of:\[A/x + (Bx+C)/(x^2 - 2x + 4)\],where A, B, C are constants to find.
02

Combine the fractions to obtain an equation

Combine the two fractions from the form obtained: \(A/x + (Bx+C)/(x^2 - 2x + 4)\), to make it equal to the original rational function \((4x^2 + 3x + 14) / (x^3 - 8)\). This results in:\[4x^2 + 3x + 14 = A(x^2 - 2x + 4) + Bx + C \]This equation must hold for all values of x. Therefore, we can choose appropriate values of x to solve for A, B, and C.
03

Solve for the constants A, B, and C

To calculate the values of the constants A, B, and C, we can substitute various values of \(x\) into the equation. Choosing zero for \(x\) eliminates the terms \(Bx\) in the equation and simplifies it. Substituting \(x=0\) results in: 14 = 4A + C. Next, substitute any other convenient value for \(x\). Let's say \(x=1\): \[4 * 1^2 + 3 * 1 + 14 = A * (1^2 - 2 * 1 + 4) + B * 1 + C. \]Substituting \(x=2\) solves the quadratic:\[4 * 2^2 + 3 * 2 + 14 = A * (2^2 - 2*2 + 4) + B * 2 + C.\]This gives us three simultaneous equations to solve for A, B, and C.
04

Solve the simultaneous equations for A, B, and C

Now solve the simultaneous equations to find the values of A, B, and C. The final step is to substitute the found constants back into the original equation, thereby obtaining the partial fraction decomposition for the given rational expression.

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

Describe how to solve a system of inequalities.

Exercises \(47-50\) describe a number of business ventures. For each exercise, a. Write the cost function, \(C\). b. Write the revenue function, \(R\) c. Determine the break-even point. Describe what this means. A company that manufactures small canoes has a fixed cost of \(\$ 18,000 .\) It costs \(\$ 20\) to produce each canoe. The selling price is \(\$ 80\) per canoe. (In solving this exercise, let \(x\) represent the number of canoes produced and sold.)

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. \(y=3 x-5\) \(21 x-35=7 y\)

When using the addition or substitution method, how can you tell if a system of linear equations has no solution? What is the relationship between the graphs of the two cquations?

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. \(3 x-2 y=-5\) \(4 x+y=8\)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. \(3 x-2 y=-5\) \(4 x+y=8\)
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Calculus

Read Explanation

Mechanics Maths

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.