Problem 7 - Calculus - AP Edition

Question

Write out the form of the partial fraction decomposition. (Do not find the numerical values of the coefficients.) $$\frac{4 x^{3}-x}{\left(x^{2}+5\right)^{2}}$$

Step-by-Step Solution

Answer

\(\frac{Ax + B}{x^2 + 5} + \frac{Cx + D}{(x^2 + 5)^2}\)

Step 1: Understanding the Denominator

The denominator is \( (x^2 + 5)^2 \), which is a repeated irreducible quadratic factor. For partial fraction decomposition, repeated irreducible quadratic factors like this require a specific setup.

Step 2: Setting Up the Decomposition

For an irreducible quadratic factor \( (x^2 + 5) \) repeated twice, the partial fraction decomposition takes the form: \[ \frac{Ax + B}{x^2 + 5} + \frac{Cx + D}{(x^2 + 5)^2} \]. This accounts for both the linear and quadratic appearance of the factor.

Step 3: Writing the Partial Fraction Decomposition

By combining the previous insights, we write the partial fraction decomposition as: \[\frac{4x^3 - x}{(x^2 + 5)^2} = \frac{Ax + B}{x^2 + 5} + \frac{Cx + D}{(x^2 + 5)^2}\]Notice that each numerator is of a degree one less than its corresponding denominator factor.

Key Concepts

Repeated Irreducible Quadratic Factors

When performing partial fraction decomposition, it's important to handle irreducible quadratic factors correctly, especially when they are repeated. The term "irreducible" applies to quadratic factors that cannot be further factored over the real numbers. In our exercise, the factor \(x^2 + 5\) is considered irreducible because it cannot be simplified into linear factors with real coefficients.

Because \((x^2 + 5)^2\) is repeated, it affects the way we express the partial fractions. For each distinct repeat, a separate fraction is created.

Specifically:

• The first fraction corresponds to the first occurrence of \(x^2 + 5\).
• The second fraction corresponds to the square, which is the second occurrence.

This setup allows us to systematically handle each part of the repeated factor in the numerator, capturing both simple and complex parts necessary for decomposition.

Numerator and Denominator Degrees

Understanding the degrees of the numerators and denominators is key in partial fraction decomposition. The general rule is that the degree of the numerator in each partial fraction should be one less than the degree of the denominator. This ensures each partial fraction is a proper fraction.

In our exercise:

• The denominator \(x^2 + 5\) is of degree 2.
• The numerator for this factor should then be a linear polynomial, \(Ax + B\), which is of degree 1.
• Repeating this for \((x^2 + 5)^2\), we again consider linear numerators, \(Cx + D\).

These numerators are chosen to be linear to match the requirement that they must be of a lower degree than their quadratic and its repeated counterpart.

Partial Fraction Setup

Setting up partial fractions is like preparing a recipe. Each step must be executed correctly to ensure the final equation aligns with the original expression.

For our given function \(\frac{4x^3 - x}{(x^2 + 5)^2}\), proper setup involves breaking it down into simple fractions.

• The first fraction \(\frac{Ax + B}{x^2 + 5}\) handles the simple appearance of \(x^2 + 5\).
• The second fraction \(\frac{Cx + D}{(x^2 + 5)^2}\) deals with the repeated appearance.

This setup acts like an outline that guides us through solving each part separately before summing them back together. This method outlines all components for a solution by isolating complexities into manageable pieces, allowing for easier coefficients solving in subsequent steps. It's an essential structure in integrating or solving related algebraic problems.