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Chapter 11: Problem 19

\(13-44=\) Find the partial fraction decomposition of the rational function. $$ \frac{4}{x^{2}-4} $$

Short Answer

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

Step by step solution

01

Identify Denominator

The denominator of the rational function is \(x^2 - 4\), which can be factored using the difference of squares formula: \(x^2 - 4 = (x-2)(x+2)\).
02

Set Up Partial Fractions

Using the factors from the denominator, express the rational function as a sum of partial fractions: \(\frac{4}{x^2 - 4} = \frac{A}{x-2} + \frac{B}{x+2}\), where \(A\) and \(B\) are constants to be determined.
03

Combine Fractions

Combine the partial fractions on the right-hand side: \(\frac{A}{x-2} + \frac{B}{x+2} = \frac{A(x+2) + B(x-2)}{(x-2)(x+2)}\).
04

Set Equal and Solve for Coefficients

Since the denominators are the same, set the numerators equal: \[ 4 = A(x+2) + B(x-2) \]Expand and equate coefficients:\[ 4 = Ax + 2A + Bx - 2B \]Combine like terms:\[ 4 = (A + B)x + (2A - 2B) \]This gives us the system:\( A + B = 0 \)\( 2A - 2B = 4 \).
05

Solve the System of Equations

Use the system of equations to find \(A\) and \(B\):1. \(A + B = 0\) implies \(B = -A\).2. Substitute \(B = -A\) into \(2A - 2B = 4\): \[ 2A - 2(-A) = 4 \] \[ 2A + 2A = 4 \] \[ 4A = 4 \] \[ A = 1 \]3. Since \(B = -A\), then \(B = -1\).
06

Write Final Partial Fraction Decomposition

Substitute the values of \(A\) and \(B\) back into the partial fraction form:\[ \frac{4}{x^2 - 4} = \frac{1}{x-2} - \frac{1}{x+2} \].

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Key Concepts

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

Difference of Squares
The concept of the **Difference of Squares** is a key algebraic identity used in simplifying expressions, especially in decomposition of fractions. In our given example, the expression in the denominator is \( x^2 - 4 \). This can be rewritten as a product using the difference of squares formula:
  • Formula: \( a^2 - b^2 = (a - b)(a + b) \)
  • Application: For \( x^2 - 4 \), consider \( a = x \) and \( b = 2 \), so \( x^2 - 4 = (x - 2)(x + 2) \)
This factorization is critical because it allows us to break down the rational function into simpler terms, making it easier to work with partial fraction decomposition.
Rational Functions
A **Rational Function** is a ratio of two polynomials. In this exercise, we work with the rational function \( \frac{4}{x^2 - 4} \). Understanding rational functions is important since they're common in calculus and algebra, helping to describe various real-world scenarios.
  • Numerator: The polynomial at the top, which in this case is 4.
  • Denominator: The polynomial at the bottom, \( x^2 - 4 \), which we've already factored.
The goal in partial fraction decomposition is to express the rational function as a sum of simpler fractions. By doing so, it opens up various mathematical techniques such as integration or solving equations, which would otherwise be challenging to address directly with more complex rational functions.
Algebraic Manipulation
Algebraic manipulation involves systematically rewriting expressions to simplify or solve equations. Here, it played a crucial role in the partial fraction decomposition process.
  • Identify: Recognize how to break down the denominator into factors using the difference of squares.
  • Set Up: Express the fraction as a sum of simpler fractions: \( \frac{A}{x-2} + \frac{B}{x+2} \).
  • Combine: Merge these fractions, focusing on the numerator, resulting in \( A(x + 2) + B(x - 2) \).
  • Solve: Equate the numerator from this combined fraction to the original, \( 4 = A(x + 2) + B(x - 2) \), and solve the resulting system of equations.
Finally, substitute the solutions back to confirm the partial fractions: \( \frac{1}{x-2} - \frac{1}{x+2} \). This type of algebraic manipulation is powerful for breaking down complex algebraic expressions into more manageable parts.

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Most popular questions from this chapter

Show that $$ \left|\begin{array}{lll}{1} & {x} & {x^{2}} \\ {1} & {y} & {y^{2}} \\ {1} & {z} & {z^{2}}\end{array}\right|=(x-y)(y-z)(z-x) $$

Solve the system of linear equations. $$ \left\\{\begin{array}{rr}{x+y+2 z-w=} & {-2} \\ {3 y+z+2 w=} & {2} \\ {x+y} & {+3 w=2} \\ {-3 x} & {+z+2 w=} & {5}\end{array}\right. $$

Furniture Manufacturing \(\quad\) A man and his daughter manufacture unfinished tables and chairs. Each table requires 3 hours of sawing and 1 hour of assembly. Each chair requires 2 hours of sawing and 2 hours of assembly. Between the two of them, they can put in up to 12 hours of sawing and 8 hours of assembly work each day. Find a system of inequalities that describes all possible combinations of tables and chairs that they can make daily. Graph the solution set.

Find the determinant of the matrix. Determine whether the matrix has an inverse, but don鈥檛 calculate the inverse. $$ \left[\begin{array}{rrr}{-2} & {-\frac{3}{2}} & {\frac{1}{2}} \\ {2} & {4} & {0} \\ {\frac{1}{2}} & {2} & {1}\end{array}\right] $$

21-46 . Graph the solution of the system of inequalities. Find the coordinates of all vertices, and determine whether the solution set is bounded. $$ \left\\{\begin{aligned} x & \geq 0 \\ y & \geq 0 \\ x & \leq 5 \\ x+y & \leq 7 \end{aligned}\right. $$
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