Problem 56 - Precalculus with Limits

Question

Writing the Partial Fraction Decomposition, write the partial fraction decomposition of the rational expression. Use a graphing utility to check your result. $$\frac{x^{3}}{(x+2)^{2}(x-2)^{2}}$$

Step-by-Step Solution

Answer

The partial fraction decomposition of the rational expression \(\frac{x^{3}}{(x+2)^{2}(x-2)^{2}}\) is: \(\frac{A}{x+2} + \frac{B}{(x+2)^{2}} + \frac{C}{x-2} + \frac{D}{(x-2)^{2}}\) where A, B, C and D are constants determined from solving the equation.

Step 1: Identify the Factors

First, examine the given expression and identify the denominator. It is \((x+2)^{2}(x-2)^{2}\), which contains the factors \((x+2)\) & \((x-2)\). These factors both occur twice (due to exponent 2). So we will have four terms in our partial fraction decomposition.

Step 2: Write the Partial Fraction Decomposition Structure

With the factors identified, the next step is to set up the general form of the partial fraction decomposition. Considering that each factor occurs twice, we write this as: \(\frac{x^{3}}{(x+2)^{2}(x-2)^{2}} = \frac{A}{x+2} + \frac{B}{(x+2)^2} + \frac{C}{x-2} + \frac{D}{(x-2)^2}\), where A, B, C and D are constants we need to find.

Step 3: Clear the Fractions

Multiply through by the common denominator \((x+2)^{2}(x-2)^{2}\) to remove the fractions. This gives us: \(x^{3} = A(x+2)(x-2)^{2} + B(x-2)^{2} + C(x+2)^{2}(x-2) + D(x+2)^{2}\)

Step 4: Solve for Constants

Now, choose values for x that will simplify the equation and solve it for A, B, C, and D. Choosing x = 2 and x = -2 will simplify expressions. Solve those 2 equations separately to get two constants, then add 2 more values for x to find the missing constants. After this process, we'll have the values for A, B, C, D and our decimals.

Key Concepts

Rational Expressions

A rational expression is quite like a fraction, but with polynomials in the numerator and the denominator. In simpler terms, it's a ratio of two expressions involving variables and constants. For instance, the expression \( \frac{x^3}{(x+2)^2(x-2)^2} \) is a rational expression. Here:

• The numerator is a polynomial: \( x^3 \)
• The denominator is a product of polynomials: \( (x+2)^2(x-2)^2 \)

Rational expressions are pivotal in algebra because they can often be simplified or decomposed into easier-to-manage segments, which is exactly what happens in partial fraction decomposition. This process helps in simplifying complex expressions, making them easier to analyze and work with, especially in calculus and higher-level math.

Graphing Utility

Graphing utilities, such as graphing calculators or software, are vital tools for visualizing mathematical expressions, including rational expressions. When dealing with partial fraction decomposition, graphing utilities can help verify whether you've set up your expression correctly by providing a visual representation. They allow you to:

• Plot the original rational expression and the partial fractions separately
• Visually compare the graphs to ensure they match, indicating a correct decomposition

Using a graphing utility is a straightforward way to double-check your work. After performing the decomposition, inputting both the original expression and the resulting fractions into the utility should yield identical graphs if the decomposition was done correctly.

Constants

In partial fraction decomposition, constants are values represented by letters such as \( A \), \( B \), \( C \), and \( D \) in the decomposed expression. These constants play a critical role and are determined by solving equations derived from the original rational expression. The process typically involves:

• Setting up an equation involving the original numerator equal to the expanded form of your partial fractions
• Selecting strategic values for the variable \( x \) to simplify the equation and isolate constants
• Solving for each constant to complete the decomposition

For example, setting \( x \) to values like \(-2\) and \(2\) helped simplify the expression and solve for some constants directly, streamlining the process.

Factors in Denominators

The factors in the denominator of a rational expression guide the structure of the partial fraction decomposition. In our example, the denominator \( (x+2)^2(x-2)^2 \) determines how the decomposition begins. Multiple rules apply:

• Each linear factor such as \( (x+2) \) and \( (x-2) \) leads to terms like \( \frac{A}{x+2} \) and \( \frac{C}{x-2} \)
• When factors are raised to a power, as \( (x+2)^2 \) and \( (x-2)^2 \), the decomposition must include terms for each power up to the highest degree, which are \( \frac{B}{(x+2)^2} \) and \( \frac{D}{(x-2)^2} \)

The goal is to break down the expression in such a way that each fraction has a simpler denominator, making it easier to solve for each constant. Understanding the factor structure is key to setting up an accurate and complete decomposition.