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Chapter 12: Problem 23

(a) Find an appropriate viewing rectangle to demonstrate that the following purported partial fraction decomposition is incorrect: $$\frac{2 x+5}{(x-4)(x+3)}=\frac{13 / 7}{x-4}+\frac{2 / 7}{x+3}$$ (b) Follow part (a) using $$\frac{2 x+5}{(x-4)(x+3)}=\frac{13 / 7}{x-4}-\frac{1 / 7}{x+3}$$ (c) Determine the correct partial fraction decomposition. given that it has the general form $$\frac{2 x+5}{(x-4)(x+3)}=\frac{A}{x-4}+\frac{B}{x+3}$$

Short Answer

Expert verified
The correct decomposition is \( \frac{13/7}{x-4} + \frac{1/7}{x+3} \).

Step by step solution

01

Analyze the Given Partial Fraction Decomposition

The goal of this part is to confirm that the given equation \( \frac{2x+5}{(x-4)(x+3)} = \frac{13/7}{x-4} + \frac{2/7}{x+3} \) is incorrect. We'll start by finding a viewing rectangle for the graphs of both sides of the equation to see if they match.
02

Graphing to Disprove the Equation

Plot the function \( \frac{2x+5}{(x-4)(x+3)} \) and \( \frac{13/7}{x-4} + \frac{2/7}{x+3} \). Use a viewing rectangle around the asymptotes at \( x = 4 \) and \( x = -3 \). The intersection points or lack of alignment in these regions will show if they are equivalent.
03

Analyze the Corrected Partial Fraction

Now, for part (b), we will plot \( \frac{2x+5}{(x-4)(x+3)} \) against \( \frac{13/7}{x-4} - \frac{1/7}{x+3} \). This is another attempt to verify whether the corrected fraction decomposition is right by observing graph alignment in the same rectangle.
04

Determine the Correct Partial Fraction From

Assume the form \( \frac{2x+5}{(x-4)(x+3)} = \frac{A}{x-4} + \frac{B}{x+3} \) and clear fractions by multiplying through by \((x-4)(x+3)\) to get: \[ 2x + 5 = A(x+3) + B(x-4) \]Distribute and gather like terms:\[ 2x + 5 = (A+B)x + (3A - 4B) \]. Equate the coefficients: 1. A + B = 2 2. 3A - 4B = 5 .
05

Solve for A and B

Use the coefficient equations:1. \( A + B = 2 \)2. \( 3A - 4B = 5 \)Solve the system of equations. From equation 1: \( A = 2 - B \). Substitute in equation 2:\[ 3(2-B) - 4B = 5 \]\[ 6 - 3B - 4B = 5 \]\[ 6 - 7B = 5 \]\[ -7B = -1 \]\[ B = \frac{1}{7} \].Substitute \( B = \frac{1}{7} \) back into \( A = 2 - B \):\[ A = 2 - \frac{1}{7} = \frac{14}{7} - \frac{1}{7} = \frac{13}{7} \].
06

Conclusion

The correct partial fraction decomposition is \( \frac{2x+5}{(x-4)(x+3)} = \frac{13/7}{x-4} + \frac{1/7}{x+3} \). The errors in the previous decompositions were inappropriately adjusting the coefficient of \( x+3 \) in the denominator.

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Key Concepts

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

Rational Functions
Rational functions are a type of function that is expressed as the ratio of two polynomials. In mathematical terms, it takes the form \( \frac{P(x)}{Q(x)} \) where \(P(x)\) and \(Q(x)\) are polynomials and \(Q(x)\) is not zero. These functions are intriguing because they can behave quite differently from many more common functions like linear or quadratic ones. The behavior of a rational function is largely determined by the degree of the polynomials in both the numerator and the denominator.
  • A rational function has potential for discontinuities, points where it does not have a defined value. This typically happens when the denominator becomes zero.
  • They often have asymptotes, lines that the graph approaches but never touches, which can be horizontal, vertical, or oblique.
  • They can also sometimes simplify into other types of functions if parts of the numerator and denominator cancel each other out.
It is important to familiarize oneself with their graphs and characteristics, as they play a significant role in calculus and algebra.
Asymptotes
Asymptotes are lines that a graph approaches but never quite reaches. In the context of rational functions, they reveal important characteristics about the graph's behavior.
  • Vertical asymptotes occur when the denominator of a rational function is zero and the numerator isn't. This is observed as the graph shooting up to infinity or plunging down to negative infinity.
  • Horizontal asymptotes are found by examining the degrees of the polynomials in the numerator and the denominator. If they are the same, the horizontal asymptote is \( y = \frac{a_n}{b_n} \) where \(a_n\) and \(b_n\) are the leading coefficients.
  • Oblique, or slant, asymptotes arise when the degree of the numerator is greater than the degree of the denominator by one; they can be found through polynomial long division.
Asymptotes guide how the graph behaves at extreme values, and are crucial in graphing rational functions accurately.
Graphing Techniques
Graphing rational functions involves identifying several key features, including intercepts, asymptotes, and the overall shape. A systematic approach can make the process more manageable and improve accuracy.
  • Start by finding the intercepts: solve for when the function equals zero or when the numerator equals zero to find the x-intercepts, and set \(x = 0\) to find the y-intercept.
  • Determine the asymptotes by checking the zeros of the denominator for vertical asymptotes, and compare the degree of the numerator and denominator for horizontal or oblique asymptotes.
  • Plot additional points as needed to ensure the general shape of the curve is captured, particularly around the vertical asymptotes where the behavior can change dramatically.
The correct visualization often requires a suitable viewing rectangle, which encompasses all these elements and provides a clear view of the function鈥檚 behavior, especially around discontinuities.
System of Equations
A system of equations involves solving for multiple variables simultaneously, where each equation is representative of a constraint or relation between those variables.
  • Systems may be linear, comprising equations of straight lines, or nonlinear, featuring curves and more complex shapes.
  • To solve, you can use various methods like substitution, elimination, or matrix approaches, depending on the system's complexity.
  • In the context of partial fraction decomposition, systems of equations is crucial. By equating coefficients of like terms, you can determine values for unknown constants in your decomposition equations.
Understanding how to manipulate and solve systems is fundamental for many areas of algebra and forms the basis for techniques used in linear algebra and calculus. By rewriting expressions and solving equations in this way, decomposition into simpler forms becomes possible, aiding in both solving equations and integration tasks.

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Most popular questions from this chapter

You are given an improper rational expression. First, use long division to rewrite the expression in the form (polynomial) \(+\) (proper rational expression) Next, obtain the partial fraction decomposition for the proper rational expression. Finally, rewrite the given improper rational expression in the form (polynomial) \(+\) (partial fractions) $$\frac{x^{6}+3 x^{5}+9 x^{3}+26 x^{2}+3 x+8}{x^{3}+8}$$

Determine the partial fraction decomposition for each of the given rational expressions. Hint: In Exercises \(17,18,\) and \(26,\) use the rational roots theorem to factor the denominator. $$\frac{1}{x^{3}-1}$$

You are given an improper rational expression. First, use long division to rewrite the expression in the form (polynomial) \(+\) (proper rational expression) Next, obtain the partial fraction decomposition for the proper rational expression. Finally, rewrite the given improper rational expression in the form (polynomial) \(+\) (partial fractions) $$\frac{2 x^{5}-11 x^{4}-4 x^{3}+53 x^{2}-24 x-5}{2 x^{3}+x^{2}-10 x-5}$$

(a) Determine the general form for the partial fraction decomposition of \(1 /\left(x^{4}-x^{3}+x^{2}-x+1\right) .\) (Note that you are not required to find the numerical values for each constant.) Hint: See Exercise 27 in Section 12.7 (b) Determine the general form for the partial fraction decomposition of \(1 /\left(x^{5}-1\right)\)

$$\frac{x^{5}-1}{\left(x^{2}+1\right)^{3}}$$
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