Problem 5 - Precalculus: Mathematics for Calculus

Question

Write the form of the partial fraction decomposition of the function (as in Example 4 ). Do not determine the numerical values of the coefficients. $$\frac{x^{2}-3 x+5}{(x-2)^{2}(x+4)}$$

Step-by-Step Solution

Answer

\( \frac{A}{x-2} + \frac{B}{(x-2)^2} + \frac{C}{x+4} \)

Step 1: Identify the Denominator

The denominator of the given rational function is \((x-2)^2(x+4)\). Notice that \((x-2)\) is a repeated linear factor, appearing twice, and \((x+4)\) is a distinct linear factor.

Step 2: Write the Partial Fraction Decomposition Form

For a repeated linear factor \((x-2)^2\), use the terms \(\frac{A}{x-2}\) and \(\frac{B}{(x-2)^2}\). For the distinct linear factor \((x+4)\), use the term \(\frac{C}{x+4}\). Combine these terms to form the decomposition: \[ \frac{x^2 - 3x + 5}{(x-2)^2(x+4)} = \frac{A}{x-2} + \frac{B}{(x-2)^2} + \frac{C}{x+4}. \]

Key Concepts

Rational Functions

Rational functions are core elements in algebra and calculus. A rational function is essentially a fraction where both the numerator and the denominator are polynomials. These functions can be expressed in the general form \( \frac{P(x)}{Q(x)} \), where \(P(x)\) and \(Q(x)\) are polynomials, and \(Q(x) eq 0\).

Some key aspects of rational functions include:

• The degree: The degree of a polynomial is the highest power of the variable in the expression. These degrees help determine the behavior and characteristics of the rational function.
• The domain: Since division by zero is undefined, the domain of a rational function excludes the values of \(x\) that make \(Q(x) = 0\).
• Asymptotes: These are lines that the graph of a function approaches but never touches. A rational function can have both vertical and horizontal asymptotes.

Partial fraction decomposition involves breaking down complex rational functions into simpler fractions, making them easier to integrate or analyze. This is especially useful when the denominator consists of linear factors or irreducible quadratic factors. Understanding the structure of rational functions is the first step in mastering the art of partial fraction decomposition.

Repeated Linear Factors

Repeated linear factors are unique in partial fraction decomposition. They occur when a polynomial in the denominator is raised to a power greater than one. In our exercise, \((x-2)\) is raised to the second power, making it a repeated factor.

When dealing with repeated linear factors, each instance of the factor must be considered.

• The factors are decomposed into partial fractions such as \(\frac{A}{x-2}\) for the simple occurrence and \(\frac{B}{(x-2)^2}\) for the second occurrence in our sample exercise.
• This method ensures that each power of the factor is accounted for, which is critical when solving for the coefficients in these fractions later on.

The introduction of separate terms for each power helps capture the behavior of the rational function fully. It ensures that the resulting function's characteristics align accurately with the original, allowing for a more comprehensive solution.

Distinct Linear Factors

Distinct linear factors arise when the terms in the polynomial are not repeated. Each distinct linear factor, appearing only once, contributes a simple partial fraction to the decomposition.

In the case of our exercise, \((x+4)\) is considered a distinct linear factor.

For each distinct linear factor, the partial fraction form \(\frac{C}{x+4}\) is used. This reflects:

• The need to account for singular linear elements in the denominator, ensuring each is represented separately.
• The simplification it provides in breaking down the complexity of a polynomial into distinct, manageable parts.

Addressing distinct linear factors individually ensures that partial fractions accurately reflect the original rational function's composition. This step is crucial in both analytical solutions and in practical applications like integration. Combined with understanding repeated factors, it completes the foundation necessary to manipulate and solve rational functions using partial fraction decomposition.