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Chapter 8: Problem 33

write the partial fraction decomposition of each rational expression. $$\frac{x+4}{x^{2}\left(x^{2}+4\right)}$$

Short Answer

Expert verified
The partial fraction decomposition of the given rational expression \(\frac{x+4}{x^{2}(x^2+4)}\) is \(\frac{x+4}{x^2 +4}\).

Step by step solution

01

Find Factors of Denominator

The given expression is \(\frac{x+4}{x^{2}(x^2+4)}\), where the denominator is already factored. There are 2 factors, one quadratic \(x^2\) and one linear \((x^2+4)\). The type of the factor will determine the form of the fractions in the decomposition i.e. linear or quadratic.
02

Form Partial Fractions

The next step is to form fractions corresponding to each factor. The partial fraction decomposition of the rational expression will be in the form \(\frac{A}{x} + \frac{B}{x} + \frac{Cx+D}{x^2+4}\), where A, B, C, and D are constants to be found.
03

Solve for Constants

To solve for the constants, first equate the given rational expression with the partial fractions and then arrange the equation in a form where the coefficients can be easily found by equating them with the coefficients of the original expression. This results in the equation \(x+4 = Ax^2 +Bx^2 + Cx^3 + Dx.\) Equate the powers and then solve for A, B, C, and D. Using the coefficients of the RHS, we can deduce that A + B = 0, and C=1, D = 4.
04

Substitute Constants into Partial Fractions

Substitute the values of the constants found in the previous step into the partial fractions. The final partial fraction decomposition of the given rational expression will be \(\frac{0}{x} + \frac{0}{x} + \frac{x+4}{x^2 +4}\), which simplifies to \(\frac{x+4}{x^2 +4}\).

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Key Concepts

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

Rational Expressions
Rational expressions are mathematical fractions where both the numerator and the denominator are polynomials. They play a crucial role in algebra and calculus. Just like regular fractions, they need to be simplified by factoring both parts to see what can be canceled out. In this context, decomposition is vital to break a complex rational expression into simpler parts. This makes it easier to integrate or differentiate them. An example here is the rational expression \( \frac{x+4}{x^2(x^2+4)} \).
  • The numerator is \( x+4 \), a simple polynomial.
  • The denominator is \( x^2(x^2+4) \), already factored into quadratic and polynomial parts.
Understanding rational expressions helps in identifying the factors and determining how to decompose them using partial fractions.
Quadratic Factors
Quadratic factors are expressions of the form \( ax^2 + bx + c \). In partial fraction decomposition, these types of factors determine the form of one of the terms in the decomposition. For instance, if you have a quadratic factor like \( x^2 + 4 \), as in our exercise, it can't be factored further using real numbers.
When dealing with quadratic factors in partial fraction decomposition:
  • The term for the factor \( x^2 + 4 \) should be a linear expression \( Cx + D \).
  • The goal is to make sure you account for its complexity compared to simpler linear factors.
Setting up your partial fraction like this ensures you can solve for the constants correctly.
Linear Factors
Linear factors are simpler polynomial expressions of the form \( ax + b \). When decomposing a rational expression, linear factors guide the setup of simpler partial fraction terms. In our problem, \( x^2 \) can be viewed as repeated linear factor \( (x)(x) \).
Why repeat the linear factor in decomposition?
  • Each factor represents a separate term in the partial fraction.
  • They use constants (A, B, etc.) in the numerators reflecting the factor's simplicity.
Remember, when dealing with repeated linear factors like \( x^2 \), use terms like \( \frac{A}{x} + \frac{B}{x} \), ensuring you address each instance of the factor.
Constants
Constants are crucial in partial fraction decomposition because they form the solution components. These constants are represented by A, B, C, and D in the decomposed terms. They're numbers needed to satisfy the equality between the original expression and its partial fraction form.
Steps to Determine Constants:
  • Set the original expression equal to the sum of your partial fractions.
  • Expand and equate coefficients for corresponding powers of x.
  • Solve the resulting system of equations to find the values of A, B, C, and D.
In our exercise, we found C=1, D=4, while A and B were interrelated by \( A + B = 0 \). This systematic approach ensures the decomposition is accurate and reflects the properties of the original rational expression.

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Most popular questions from this chapter

What is a system of linear equations? Provide an example with your description.

determine whether each statement makes sense or does not make sense, and explain your reasoning. Use an extension of the Great Question! on page 859 to describe how to set up the partial fraction decomposition of a rational expression that contains powers of a prime cubic factor in the denominator. Give an example of such a decomposition.

Exercises \(41-43\) will help you prepare for the material covered in the first section of the next chapter. Solve the system: $$ \left\\{\begin{aligned} x+y+2 z &=19 \\ y+2 z &=13 \\ z &=5 \end{aligned}\right. $$

a. A student earns \(\$ 15\) per hour for tutoring and \(\$ 10\) per hour as a teacher's aide. Let \(x=\) the number of hours each week spent tutoring and let \(y=\) the number of hours each week spent as a teacher's aide. Write the objective function that models total weekly earnings. b. The student is bound by the following constraints: \(\cdot\) To have enough time for studies, the student can work no more than 20 hours per week. \(\cdot\) The tutoring center requires that each tutor spend at least three hours per week tutoring. \(\cdot\) The tutoring center requires that each tutor spend no more than eight hours per week tutoring. Write a system of three inequalities that models these constraints. c. Graph the system of inequalities in part (b). Use only the first quadrant and its boundary, because \(x\) and \(y\) are nonnegative. d. Evaluate the objective function for total weekly earnings at each of the four vertices of the graphed region. [The vertices should occur at \((3,0),(8,0),(3,17),\) and \((8,12) .]\) e. Complete the missing portions of this statement: The student can earn the maximum amount per week by tutoring for ____ hours per week and working as a teacher's aide for ____ hours per week. The maximum amount that the student can earn each week is $_____.

If \(x=3, y=2,\) and \(z=-3,\) does the ordered triple \((x, y, z)\) satisfy the equation \(2 x-y+4 z=-8 ?\)
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