Problem 5 - Algebra and Trigonometry

Question

\(3-12\) . 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

The partial fraction form is \( \frac{A}{x-2} + \frac{B}{(x-2)^2} + \frac{C}{x+4} \).

Step 1: Identify the Denominator Factors

First, observe the denominator \((x-2)^2(x+4)\). It consists of two distinct linear factors \((x - 2)\) and \((x + 4)\), where \((x - 2)\) is repeated.

Step 2: Set Up Partial Fraction Form

For a denominator like \((x-2)^2(x+4)\), the partial fraction decomposition consists of a term for each power of the repeated factor and one for the distinct linear factor. Therefore, the decomposition will be: \[ \frac{A}{x-2} + \frac{B}{(x-2)^{2}} + \frac{C}{x+4} \].

Step 3: Write the Partial Fraction Form

Based on the previous analysis, the partial fraction form of \( \frac{x^{2}-3x+5}{(x-2)^{2}(x+4)} \) is given by: \[ \frac{A}{x-2} + \frac{B}{(x-2)^2} + \frac{C}{x+4} \], where \( A \), \( B \), and \( C \) are the coefficients to be determined.

Key Concepts

Rational Expressions

Rational expressions are like fractions where both the numerator and denominator are polynomials. These are quite common in algebra and calculus. In the expression given in the exercise, \( \frac{x^{2}-3x+5}{(x-2)^{2}(x+4)} \), the polynomial on top is \( x^{2}-3x+5 \) and on the bottom are the polynomials \( (x-2)^{2}(x+4) \). Understanding these expressions is essential because they often appear in mathematical problems, and simplifying or decomposing them can make equations easier to solve. Rational expressions may look complex at first, but breaking them down into smaller parts can simplify the problem.

Denominator Factors

The denominator of a rational expression contains factors which are crucial to solving it through partial fraction decomposition. In the given example, the denominator \((x-2)^{2}(x+4)\) consists of two factors: \((x-2)^2\) and \((x+4)\).Denominator factors determine how the partial fraction decomposition will be set up. Identifying these clearly is the first step in properly decomposing the rational expression. Understanding the role of each factor, especially noticing which ones repeat, is a critical skill when working with rational expressions in algebra.

Linear Factors

Linear factors are factors of the form \( ax + b \), which represent straight lines when plotted on a graph. In our problem, \( (x - 2) \) and \( (x + 4) \) are linear factors because they can be written in the form \( ax + b \), where \( a = 1 \) and \( b = -2 \) or \( 4 \) respectively. In partial fraction decomposition, each linear factor will have a corresponding term in the decomposition. For a simple linear factor \( (x - 2) \), you would have a term like \( \frac{A}{x-2} \). This approach simplifies the process of solving integrals or simplifying complex algebraic expressions.

Repeated Factors

Repeated factors occur when a factor appears more than once in the denominator of a rational expression. They are particularly important in partial fraction decomposition. In this exercise, \((x-2)\) is a repeated factor, since it appears squared, as \((x-2)^2\).When decomposing, each different power of a repeated factor requires its own term in the decomposition. Therefore, due to \((x-2)^2\) in the denominator, we have terms \( \frac{A}{x-2} \) and \( \frac{B}{(x-2)^2} \). Each of these parts is essential to accurately represent the original expression, providing a simpler way to work with the equation.