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Chapter 5: Problem 47

Write the partial fraction decomposition of each rational expression. $$\frac{1}{x^{2}-c^{2}} \quad(c \neq 0)$$

Short Answer

Expert verified
\(\frac{1}{x^2 - c^2} = \frac{1/2c}{x - c} - \frac{1/2c}{x + c}\).

Step by step solution

01

Identify the Difference of Squares

The denominator is a difference of squares, and can be factored as \(x^2 - c^2 = (x - c)(x + c)\) by using the difference of squares formula.
02

Write out the Partial Fraction Decomposition

The concept of partial fraction decomposition suggests that a complex fraction can be decomposed as sum of simpler fractions. Here, the rational expression can be written as: \(\frac{1}{x^2-c^2} = \frac{A}{x - c} + \frac{B}{x + c}\) where A and B are constants to be found.
03

Resolve the Constants A and B

Multiply both sides by the common denominator \(x^2 - c^2\) to remove the fractions and simplify: \(1 = A(x + c) + B(x - c)\). To solve for A and B, two convenient choices for \(x\) would be \(c\) and \(-c\). When \(x = c\), the equation simplifies to \(1 = 2Ac\), so \(A = \frac{1}{2c}\). When \(x = -c\), the equation simplifies to \(1 = -2Bc\), so \(B = -\frac{1}{2c}\).
04

Write the Final Answer

Substitute A and B back into the partial fraction decomposition: \(\frac{1}{x^2 - c^2} = \frac{1/2c}{x - c} - \frac{1/2c}{x + c}\).

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Key Concepts

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

Difference of Squares
Understanding the difference of squares is fundamental to solving various algebraic problems, particularly when dealing with factorization. It refers to an expression of the form \(a^2 - b^2\), which can be factored into \(a + b)(a - b)\). For example, if we apply the difference of squares to \(x^2 - c^2\), we get \(x - c)(x + c)\).

This property is very useful as it turns a seemingly complex expression into a product of two first-degree binomials, which are often easier to work with, especially when involved in rational expressions that require decomposition, such as in the partial fraction decomposition exercise.
Rational Expression
A rational expression is a fraction where both the numerator and the denominator are polynomials. In the exercise, \(\frac{1}{x^2-c^2}\) is a rational expression, with the numerator being 1 (a constant polynomial), and the denominator being \(x^2-c^2\), a polynomial of degree 2. Simplifying rational expressions often involves factorization of these polynomials, where techniques like the difference of squares come into play. Recognizing rational expressions and their properties is key to manipulating and solving algebraic fractions.
Algebraic Fractions
Algebraic fractions, also known as fractional expressions, have variables in the numerator, denominator, or both. The process of breaking them into simpler parts, or decomposing them, is essential for simplification or integration of expressions. In our exercise, the algebraic fraction \(\frac{1}{x^2 - c^2}\) cannot be simplified as is, but through partial fraction decomposition, it can be expressed as a sum of simpler fractions that are easier to integrate or evaluate, thus better aiding in the computation of more complex algebraic tasks.
Factorization
Factorization is the process of breaking down a complex expression into a product of simpler factors. This is often a crucial step in solving equations, as seen in partial fraction decomposition. When a polynomial includes terms that are perfect squares separated by a minus sign, we can apply the difference of squares rule, which is a special case of factorization. In our problem, the denominator \(x^2 - c^2\) is factored into \(x - c)(x + c)\), which allows us to rewrite the rational expression into a more manageable form for further operations.

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Most popular questions from this chapter

On your next vacation, you will divide lodging between large resorts and small inns. Let \(x\) represent the number of nights spent in large resorts. Let \(y\) represent the number of nights spent in small inns. a. Write a system of inequalities that models the following conditions: You want to stay at least 5 nights. At least one night should be spent at a large resort. Large resorts average \(\$ 200\) per night and small inns average \(\$ 100\) per night. Your budget permits no more than \(\$ 700\) for lodging. b. Graph the solution set of the system of inequalities in part (a). c. Based on your graph in part (b), what is the greatest number of nights you could spend at a large resort and still stay within your budget?

Many elevators have a capacity of 2000 pounds. a. If a child averages 50 pounds and an adult 150 pounds, write an inequality that describes when \(x\) children and \(y\) adults will cause the elevator to be overloaded. b. Graph the inequality. Because \(x\) and \(y\) must be positive, limit the graph to quadrant I only. c. Select an ordered pair satisfying the inequality. What are its coordinates and what do they represent in this situation?

The figure shows the healthy weight region for various heights for people ages 35 and older. GRAPH CAN'T COPY If \(x\) represents height, in inches, and \(y\) represents weight, in pounds, the healthy weight region can be modeled by the following system of linear inequalities: $$\left\\{\begin{array}{l} 5.3 x-y \geq 180 \\ 4.1 x-y \leq 140 \end{array}\right.$$ Use this information to solve Exercises 77-80. Is a person in this age group who is 5 feet 8 inches tall weighing 135 pounds within the healthy weight region?

What is a half-plane?

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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