Problem 7 - College Algebra

Question

Write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants. $$\frac{x^{3}+x^{2}}{\left(x^{2}+4\right)^{2}}$$

Step-by-Step Solution

Answer

The form of the partial fraction decomposition of the given rational function is \( \frac{Ax + B}{x^{2}+4} + \frac{Cx + D}{(x^{2}+4)^2} \).

Step 1: Recognize the Form of Denominator

The denominator of the rational function, \( (x^{2}+4)^{2} \), is a perfect square. This suggests that the partial fractions would each contain a term of the form \( \frac{1}{(x^{2}+4)^{k}} \) where \( k \) ranges from 1 to the power of the binomial, which is 2 in this case. Therefore, the final answer will have two fractions as our denominator has a power of 2.

Step 2: Determine the Numerator for Each Term

Since the denominator consists of quadratics, the numerator has to be one degree lower than the degree of the denominator for each fraction. Since the degree of denominator is 2, the numerator should be linear (degree 1) or less. In this case, the proper form of the numerator is Ax + B where A and B are constants.

Step 3: Form the Partial Fraction Decomposition

Combine the numerator and denominator from previous steps, the partial fraction decomposition of the given rational function can be written as: \( \frac{Ax + B}{x^{2}+4} + \frac{Cx + D}{(x^{2}+4)^2} \)

Key Concepts

Rational Expressions

Rational expressions are fractions where both the numerator and the denominator are polynomials. To better understand this, think of rational expressions as the algebraic equivalent of numerical fractions. This simplification allows mathematicians and students to work with them under similar rules as regular fractions, including operations like addition, subtraction, multiplication, and division.

In the context of partial fraction decomposition, rational expressions become particularly important because the goal is to express a complex rational expression as a sum of simpler fractions. For instance, the expression \(\frac{x^{3}+x^{2}}{(x^{2}+4)^{2}}\) is a rational expression that needs to be broken down into simpler components for easier manipulation and understanding.

Polynomial Degree

The degree of a polynomial is the highest power of the variable in its expression. It's a crucial concept because it helps determine how expressions are simplified or decomposed. In our rational expression \(\frac{x^{3}+x^{2}}{(x^{2}+4)^{2}}\), the numerator is a polynomial of degree 3, and its denominator’s polynomial is a degree 4 polynomial because it is squared twice.

Understanding the degrees of polynomials involved is vital because it guides the form of the numerators in the partial fraction decomposition. Specifically, the numerators in the decomposition must be of lower degree than their corresponding denominators to maintain the standardization and solvability of the resulting expression.

Numerator and Denominator

The numerator and denominator of a rational expression play distinct roles in its structure. The numerator is the top part of a fraction, representing how many parts you have, and the denominator, as the bottom part, indicates how many equal parts the whole is divided into. For instance, in \(\frac{x^{3}+x^{2}}{(x^{2}+4)^{2}}\), \(x^{3}+x^{2}\) is the numerator, and \((x^{2}+4)^{2}\) is the denominator.

When applying partial fraction decomposition, the form of these two components needs to be specially tailored. The denominator determines the terms of the partial fractions, while the numerator shapes what the constants multipliers will be. This distinction sets the groundwork for creating a sum of fractions that accurately represents the original rational expression.

Constants in Fractions

In partial fraction decomposition, constants play a critical role in forming the numerators of the proposed fractions. These constants, often represented as A, B, C, etc., are coefficients that need to be determined to express the rational function properly as a sum of simpler fractions.

For instance, in our decomposition of \(\frac{x^{3}+x^{2}}{(x^{2}+4)^{2}}\), the constants appear in the form \(\frac{Ax + B}{x^{2}+4} + \frac{Cx + D}{(x^{2}+4)^2}\). These constants will eventually be solved through a system of equations derived from setting terms equal when matching across both sides of the equation during decomposition. While our task here doesn't involve finding these constant values, recognizing their placeholder significance is essential for understanding the decomposition process.