Question

Give the form of the partial fraction expansion for the given rational function \(F(s)\). You need not evaluate the constants in the expansion. However, if the denominator of \(F(s)\) contains irreducible quadratic factors of the form \(s^{2}+2 \alpha s+\beta^{2}, \beta^{2}>\alpha^{2}\), complete the square and rewrite this factor in the form \((s+\alpha)^{2}+\omega^{2}\). $$F(s)=\frac{s^{3}-1}{\left(s^{2}+1\right)^{2}(s+4)^{2}}$$

Short answer

Question: Rewrite the given rational function in partial fraction form: \(F(s) = \frac{N(s)}{\left(s^{2}+1\right)^{2}(s+4)^{2}}\) Answer: \(F(s) = \frac{A_1}{s^2+1} + \frac{A_2}{(s^2+1)^2} + \frac{B_1}{s+4} + \frac{B_2}{(s+4)^2}\)

Step-by-step solution

Identify each factor in the denominator

The given function has denominator \(D(s) = \left(s^{2}+1\right)^{2}(s+4)^{2}\). We can split this denominator into its factors: 1. Repeated irreducible quadratic factor: \((s^2 +1)^2\) 2. Repeated linear factor: \((s+4)^2\)

Set up partial fraction form

To expand \(F(s)\) using partial fractions, we need to write it as a sum of simpler fractions with each denominator being a factor of the original denominator. Since \((s^2+1)\) and \((s+4)\) are both repeated factors, we need to include their powers in the partial fraction expansion. Therefore, we can write: $$F(s) = \frac{A_1}{s^2+1} + \frac{A_2}{(s^2+1)^2} + \frac{B_1}{s+4} + \frac{B_2}{(s+4)^2}$$

Complete the square

Here, the denominator \((s^2 + 1)\) is already in the form \((s + \alpha)^2 + \omega^2\) with \(\alpha = 0\) and \(\omega^2 = 1\). So, there is no need to complete the square for the denominator of the given function.

Obtain the partial fraction expansion

Now, we have the partial fraction expansion for \(F(s)\): $$F(s) = \frac{A_1}{s^2+1} + \frac{A_2}{(s^2+1)^2} + \frac{B_1}{s+4} + \frac{B_2}{(s+4)^2}$$ We are not required to evaluate the constants \(A_1\), \(A_2\), \(B_1\), and \(B_2\) in the expansion. We have found the form of the partial fraction expansion as requested.

Key concepts

Rational Functions

A rational function is a fraction where both the numerator and the denominator are polynomials. These types of functions can often become complex when dealing with integrals or other calculations, so breaking them down into simpler parts helps us understand and work with them more easily. This process is known as partial fraction decomposition.
To decompose the rational function into partial fractions, we need to examine its denominator. By examining the components of the denominator, we can rewrite the function as a sum of multiple simpler fractions. This makes tasks like integration and more complex algebraic manipulations easier.
Rational functions are pivotal in both calculus and algebra because they provide a bridge to understanding various advanced mathematical concepts. They arise often in real-world problems, such as in engineering and physics, where they help model various systems.

Irreducible Quadratic Factors

Sometimes while working with rational functions, we encounter irreducible quadratic factors in the denominator. An irreducible quadratic is a polynomial of the form \[ s^2 + 2\alpha s + \beta^2 \]where it cannot be factored further using real numbers.
When dealing with these denominators, it's sometimes necessary to rewrite the quadratic in its completed square form. For example, the factor \[ s^2 + 1 \]in our exercise is already in this format: \[ (s + \alpha)^2 + \omega^2 \]where \( \alpha = 0 \)and \( \omega^2 = 1 \).
This conversion allows us to set up the correct partial fraction for each irreducible quadratic factor. It is crucial in simplifying the rational function and ensuring precise calculation steps without getting bogged down by the quadratic complexity.

Denominator Expansion

The expansion of the denominator of a rational function into partial fractions is a methodical process. We break down the function's denominator into its simplest factors, laying down the foundation for our partial fraction form.
For \( F(s) \) from the exercise, the denominator can be divided into both repeated linear and repeated irreducible quadratic factors such as \( (s^2 + 1)^2 \)and \( (s + 4)^2 \). Each factor contributes separate "pieces" to the partial fractions, allowing expression of the whole as a sum of these separate parts.
The process involves setting up separate fractions for each power of the factors observed. In our problem, since both factors are repeated, we introduce multiple terms for each, as shown in \( \frac{A_1}{s^2+1} + \frac{A_2}{(s^2+1)^2} + \frac{B_1}{s+4} + \frac{B_2}{(s+4)^2} \).
Understanding denominator expansion helps in tackling complex problems by simplifying the structure step-by-step.