Problem 5 - Precalculus: Mathematical 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. $$\frac{x^{2}}{(x-3)\left(x^{2}+4\right)}$$

Step-by-Step Solution

Answer

The partial fraction form is \(\frac{A}{x-3} + \frac{Bx + C}{x^2 + 4}\).

Step 1: Identify the Denominator

First, observe the denominator of the function, which is \( (x - 3)(x^2 + 4) \). It is composed of a linear factor \(x-3\), and an irreducible quadratic factor \( x^2 + 4 \). This means each of these factors will be a part of our partial fraction decomposition.

Step 2: Set Up the Partial Fraction Decomposition

According to partial fraction decomposition rules, the expression will be written as a sum of fractions where each fraction has one of the factors from the denominator. For the linear factor \( x - 3 \), the numerator will be a constant term \( A \). For the quadratic factor \( x^2 + 4 \), the numerator will be a linear term \( Bx + C \). Thus, the partial fraction decomposition is \[ \frac{x^2}{(x - 3)(x^2 + 4)} = \frac{A}{x-3} + \frac{Bx + C}{x^2 + 4} \].

Key Concepts

Partial Fractions

Partial fraction decomposition is a technique used to express a rational function as a sum of simpler fractions, called partial fractions. This method is particularly useful for integration and simplifying complex expressions. The process involves breaking down a complex fraction into manageable pieces that are easier to work with. To start, we look at the original function to decompose.Consider the function given in the problem: \[ \frac{x^{2}}{(x-3)(x^{2}+4)} \]This represents a rational function where the degree of the polynomial in the numerator is less than the degree of the polynomial in the denominator. By decomposing it into partial fractions, we aim to simplify it into a sum format, which can be more beneficial when performing integration tasks or solving equations involving the expression.

Linear Factors

Linear factors are polynomial expressions where the variable is raised to the first power, such as \(x - 3\). These are the simplest type of factors you can encounter in a polynomial denominator. When a linear factor appears in the denominator, it contributes to a simple partial fraction with a constant numerator.In our problem, the linear factor is \(x - 3\). We assign a constant \(A\) as the numerator for this partial fraction. So, a part of our decomposition of the function becomes:\[ \frac{A}{x-3} \]Linear factors are relatively straightforward in decompositions, as they do not require more complex terms in the numerator beyond a single constant.

Quadratic Factors

Quadratic factors such as \(x^2 + 4\) are polynomial terms where the variable is raised to the second power and cannot be simplified into linear factors over the real numbers. These are more complex than linear factors and require different treatment in partial fraction decomposition.For an irreducible quadratic factor like \(x^2 + 4\), the numerator is not just a constant; instead, it needs to be a linear expression to ensure the degrees of freedom necessary for solving the coefficients. Thus, our numerator becomes \(Bx + C\) for such quadratic terms.In this way, the decomposed form of the integral part is:\[ \frac{Bx + C}{x^2 + 4} \]Handling quadratic factors appropriately by introducing linear numerators allows the proper setup for eventual solving of specific coefficients if needed later on. This step is crucial to ensure that partial fractions are accurately represented.