Problem 10 - 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. $$\frac{x^{4}+x^{2}+1}{x^{2}\left(x^{2}+4\right)^{2}}$$

Step-by-Step Solution

Answer

The partial fraction decomposition is given by: \( \frac{A}{x} + \frac{B}{x^2} + \frac{Cx+D}{x^2+4} + \frac{Ex+F}{(x^2+4)^2} \).

Step 1: Identify the Denominator Factors

The denominator of the function is given by \( x^2 (x^2 + 4)^2 \). There are two types of factors here: the linear factor \( x^2 \), and the irreducible quadratic factor \( (x^2 + 4) \), raised to the second power.

Step 2: Set Up Partial Fractions for Linear Factors

For the linear factor \( x^2 \), since it's repeated (\( x\) occurs twice because it’s squared), write partial fractions for each power up to 2 as \( \frac{A}{x} + \frac{B}{x^2} \).

Step 3: Set Up Partial Fractions for Irreducible Quadratic Factors

For the irreducible quadratic factor \( (x^2 + 4)^2 \), write terms for each power of the factor: \( \frac{Cx + D}{x^2 + 4} + \frac{Ex + F}{(x^2 + 4)^2} \). The coefficients \( C, D, E, \) and \( F \) correspond to the numerators that involve linear terms in \( x \).

Step 4: Combined Form of the Partial Fraction Decomposition

Combine all parts to write the decomposition as: \[ \frac{x^{4}+x^{2}+1}{x^{2}(x^{2}+4)^{2}} = \frac{A}{x} + \frac{B}{x^2} + \frac{Cx + D}{x^2 + 4} + \frac{Ex + F}{(x^2 + 4)^2} \]. Note: The task specifies that you do not need to find the values of the coefficients \( A, B, C, D, E, \) and \( F \).

Key Concepts

Irreducible Quadratic Factors

In partial fraction decomposition, the term "irreducible quadratic factors" refers to quadratic expressions in the denominator that cannot be factored into real linear factors. This implies that such expressions do not have real roots and remain in the quadratic form. In the provided exercise, \( (x^2 + 4) \) serves as an irreducible quadratic factor since it cannot be factorized further using real numbers.

When dealing with irreducible quadratic factors in partial fraction decomposition, linear expressions of the form \( Cx + D \) are placed in the numerators. The reasoning is that this accounts for the possibility of any real solution or partial version that the numerator can take on. It's essential to remember that for every power of the irreducible quadratic factor, a new partial fraction term needs to be included. So for \( (x^2 + 4)^2 \), two terms are written, one with \( (x^2 + 4) \) and the other with \( (x^2 + 4)^2 \).

• When it cannot be factored further in the reals, it stays quadratic.
• Use a linear expression (\( Cx + D \)) in the numerator for each power.
• Include as many terms as the factor's power indicates.

Linear Factors

Linear factors in partial fraction decomposition are simpler elements of the polynomial that can be expressed in the form \( x - a \). In this exercise, the factor \( x^2 \) is regarded as a repeated linear factor since it's the square of the linear factor \( x \).

Understanding linear factors and their involvement in decomposition is crucial. Since \( x^2 \) is squared, the decomposition must include terms for each power of the factor up to its exponent in the denominator. This results in writing a decomposition involving \( \frac{A}{x} + \frac{B}{x^2} \). By setting up partial fractions in this manner, each term corresponds to a solution accounting for all roots or potential shades of solutions related to the factor.

• Linear factors are expressed as \( x - a \).
• For repeated factors like \( x^2 \), use terms for each instance (\( \frac{A}{x} + \frac{B}{x^2} \)).
• Ensure all powers are represented.

Rational Functions

Rational functions are expressions that consist of a numerator and a denominator, where both are polynomials. The form is generally expressed as \( \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) stand for the polynomial functions in the numerator and the denominator, respectively. The partial fraction decomposition, as seen in the exercise, involves breaking down complex rational functions into simpler components.

By understanding the factoring of the denominator in a rational function, you can determine how to set up the partial fractions. This is vital as it helps simplify integrations or other operations. Address each factor type separately—linear and irreducible quadratic—as seen in the decomposition steps.

• Rational functions have a polynomial numerator and polynomial denominator.
• Decomposition breaks them into simpler fractions for easier manipulation.
• By simplifying the rational function, complex processes become more manageable.