Problem 19 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify Denominator

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

Step 2: 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.

Step 3: Combine Fractions

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

Step 4: 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:\( A + B = 0 \)\( 2A - 2B = 4 \).

Step 5: 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 \(B = -1\).

Step 6: Write Final Partial Fraction Decomposition

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

Key Concepts

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.