/*! 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 40 In Exercises 19-42, write the pa... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 40

In Exercises 19-42, write the partial fraction decomposition of the rational expression. Check your result algebraically. \( \dfrac{3}{x^4 + x} \)

Short Answer

Expert verified
The partial fraction decomposition of \(\frac{3}{x^4 + x}\) is \(\frac{1}{x} - \frac{1}{x+1} + \frac{1}{x^2-x+1}\).

Step by step solution

01

Factor the denominator

The first step to partial fraction decomposition involves factoring the denominator. The denominator is \(x^4 + x\). This can be factored by removing an x out of each term, resulting in \(x(x^3 + 1)\). But there's still more factoring to be done, as \(x^3 + 1\) can be factored using sum of cubes formula to \( (x+1)(x^2-x+1)\). This gives the final factored form of the denominator as \(x(x+1)(x^2-x+1)\).
02

Break down into partial fractions

The next step is to decompose the given fraction to partial fractions. Our factored denominator consists of 3 factors, two of which are linear, \(x\) and \(x+1\), and one which is quadratic, \(x^2-x+1\). So, we can write our fraction as \(\frac{3}{x(x+1)(x^2-x+1)} = \frac{A}{x} + \frac{B}{x+1} + \frac{Cx+D}{x^2-x+1}\), where A, B, C and D are constants that we need to find out.
03

Solve for constants

Next, we clear the fractions and simplify the equation by multiplying each side of the equation by the factored denominator \(x(x+1)(x^2-x+1)\). After equating coefficients with the original rational function \(\frac{3}{x^4 + x}\), we can solve for constants A, B, C and D. After solving these constants, we get A=1, B=-1, C=0 and D=1.
04

Write the final partial fraction decomposition

After solving the constants, we substitute these constants into our formula from step 2 and get our final partial fraction decomposition which is: \(\frac{1}{x} - \frac{1}{x+1} + \frac{1}{x^2-x+1}\)

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!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Factoring Polynomials
Factoring polynomials is a crucial skill in algebra and essential for simplifying expressions and solving equations. When we factor a polynomial, we are looking for simpler polynomials that multiply together to give the original. For example, when factoring the denominator in the exercise, we look for factors of the term x4 plus x. A common technique starts with factoring out the greatest common factor (GCF), which in this case is x.

Once the GCF is factored out, we often look for patterns to further factor the polynomial. Common patterns include the difference of squares, perfect square trinomials, and sum and difference of cubes. Recognizing these patterns can vastly simplify the original expression and is crucial for decomposing fractions in to partial fractions.
Sum of Cubes
The sum of cubes is an important algebraic factoring formula represented by the equation a3 + b3 = (a + b)(a2 - ab + b2). In our exercise, it is used to factor x3 + 1.

By comparing to the formula, we see that a = x and b = 1, which results in the factored form (x + 1)(x2 - x + 1). Noticing the sum of cubes formula allows us to break down the polynomial into simpler parts that make finding a partial fraction decomposition possible. Mastery of this and similar formulas is vital for working with higher-degree polynomials.
Rational Expressions
Rational expressions are fractions that contain polynomials in the numerator, the denominator, or both. Simplifying rational expressions, just like simplifying numerical fractions, often involves factoring polynomials to cancel out common factors. However, when dealing with partial fraction decomposition like in our given exercise, we break a complex rational expression into simpler parts.

These simpler 'partial fractions' are easier to work with, whether for integration in calculus or simplifying algebraic expressions. The goal of partial fraction decomposition is to express the rational expression as a sum of simpler rational expressions, as is shown in the step-by-step solution. Understanding rational expressions and how to decompose them is fundamental for more advanced mathematical topics.
Algebraic Manipulation
Algebraic manipulation involves rearranging and simplifying expressions using algebraic properties. In the context of partial fraction decomposition, once we determine the form of the partial fractions, we use algebraic manipulation to find the constants for each fraction. This process involves multiplying both sides of the equation by the common denominator to eliminate fractions and then equating coefficients of corresponding powers of x on both sides of the equation.

Algebraic manipulation uses properties like the distributive property, combining like terms, and equating coefficients strategically to simplify expressions or solve equations. Developing the skill of algebraic manipulation is one of the most beneficial things a mathematics student can do, as it applies across virtually all areas of math.

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 7-20, sketch the graph of the inequality. \( x \ge 6 \)

An animal shelter mixes two brands of dog food. Brand \(\mathrm{X}\) costs \(\$ 25\) per bag and contains two units of nutritional element A, two units of element \(\mathrm{B},\) and two units of element \(\mathrm{C}\) , Brand \(\mathrm{Y}\) costs \(\$ 20\) per bag and contains one unit of nutritional element A, nine units of element \(\mathrm{B},\) and three units of element \(\mathrm{C}\) . The minimum required amounts of nutrients \(\mathrm{A}, \mathrm{B},\) and \(\mathrm{Care}\) 12 units, 36 units, and 24 units, respectively. What is the optimal number of bags of each brand that should be mixed? What is the optimal cost?

Two concentric circles have radii \( x \) and \( y \), where \( y > x \). The area between the circles must be at least 10 square units. (a) Find a system of inequalities describing the constraints on the circles. (b) Use a graphing utility to graph the system of inequalities in part (a). Graph the line \( y = x \) in the same viewing window. (c) Identify the graph of the line in relation to the boundary of the inequality. Explain its meaning in the context of the problem.

In Exercises 37-40, determine whether each ordered pair is a solution of the system of linear inequalities. \( \left\\{\begin{array}{l} 3x + y > 1\\\ -y - \dfrac{1}{2}x^2 \le -4\\\ -15x + 4y > 0\end{array}\right. \) (a) \( (0, 10) \) (b) \( (0, -1) \) (c) \( (2, 9) \) (d) \( (-1, 6) \)

An accounting firm has 780 hours of staff time and 272 hours of reviewing time available each week. The firm charges \( \$1600 \) for an audit and \( \$250 \) for a tax return. Each audit requires 60 hours of staff time and 16 hours of review time. Each tax return requires 10 hours of staff time and 4 hours of review time. What numbers of audits and tax returns will yield an optimal revenue? What is the optimal revenue?
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation

Decision Maths

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.