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 \((s+\alpha)^{2}+\omega^{2}\). $$F(s)=\frac{s^{3}+3 s+1}{(s-1)^{3}(s-2)^{2}}$$

Short answer

Question: Rewrite the following function as a partial fraction expansion: $$F(s) = \frac{s^3 + 3s + 1}{(s-1)^3 (s-2)^2}$$ Answer: The partial fraction expansion of the given function is: $$F(s) = \frac{A}{(s-1)} + \frac{B}{(s-1)^2} + \frac{C}{(s-1)^3} + \frac{D}{(s-2)} + \frac{E}{(s-2)^2}$$

Step-by-step solution

Check the denominator for irreducible quadratic factors

The given rational function is: $$F(s) = \frac{s^3 + 3s + 1}{(s-1)^3 (s-2)^2}$$ In the denominator, we have the factors \((s-1)^3\) and \((s-2)^2\). Since both of these are linear, there are no irreducible quadratic factors in the denominator. Now we can proceed to decompose the rational function into partial fractions.

Write the partial fractions decomposition

The general form of the partial fractions decomposition of a rational function with factors \((s-a)^n\) and \((s-b)^m\) is: $$F(s) = \frac{A}{(s-a)} + \frac{B}{(s-a)^2} + \frac{C}{(s-a)^3} + \frac{D}{(s-b)} + \frac{E}{(s-b)^2}$$ For our given function, this becomes: $$F(s) = \frac{A}{(s-1)} + \frac{B}{(s-1)^2} + \frac{C}{(s-1)^3} + \frac{D}{(s-2)} + \frac{E}{(s-2)^2}$$ Since the problem statement says that we don't need to find the values of these constants, our final answer is: $$F(s) = \frac{A}{(s-1)} + \frac{B}{(s-1)^2} + \frac{C}{(s-1)^3} + \frac{D}{(s-2)} + \frac{E}{(s-2)^2}$$

Key concepts

Rational Functions

A rational function is any function that can be written as the ratio of two polynomials. In mathematical terms, it takes the form \( R(x) = \frac{P(x)}{Q(x)} \), where both \( P(x) \) and \( Q(x) \) are polynomials. The rational function is defined wherever the denominator is not zero, which means \( Q(x) eq 0 \).

Rational functions are a key part of algebra and calculus because they model many real-world situations and help describe behaviors involving ratios.

They often appear in situations regarding growth and decay, circuits, rates of change, and anywhere there is a direct proportional relationship between variables.

Understanding rational functions is crucial because they lay the groundwork for more advanced topics like integration in calculus.

Linear Factors

Linear factors are components of polynomials that appear as factors of degree one, like \( (s-a) \). When dealing with rational functions, identifying these linear factors is essential for various analytical techniques, such as partial fraction decomposition.

Unlike quadratic or higher-degree polynomials, linear factors take the simplest form, which makes them straightforward to handle in mathematical calculations.

In the given rational function \( F(s) = \frac{s^3 + 3s + 1}{(s-1)^3 (s-2)^2} \), the denominator's factors, \( (s-1)^3 \) and \( (s-2)^2 \), are both linear, increased by powers.

Recognizing these linear factors is the first step in performing partial fraction decomposition, which simplifies the process of integrating rational functions.

Polynomial Long Division

Polynomial long division is a technique used to simplify rational expressions, particularly when the degree of the numerator is higher than or equal to that of the denominator. It works similarly to long division with numbers.

In the polynomial division, you divide the term with the highest power in the numerator by the highest power term in the denominator. Then, like in ordinary division, you multiply, subtract, and bring down the next term, repeating the process until you can't go any further.

Although for the given problem, polynomial long division is not needed, understanding this concept is vital because it often serves as a preliminary step before applying further techniques like partial fraction decomposition.

This method provides insight into the behavior of rational functions and can reveal simpler forms or insights about the values the function may take.

Partial Fractions Technique

The partial fractions technique is a method used to express a complex rational function as a sum of simpler fractions. This technique is invaluable when dealing with integrals of rational functions in calculus.

To perform partial fraction decomposition, you start by writing the fraction as a sum of fractions, with unknown constants in the numerators, corresponding to each factor in the denominator. In cases where the denominator includes linear factors, like in our example, it's crucial to include fractions for each power of the factor.

For the rational function \( F(s) = \frac{s^3 + 3s + 1}{(s-1)^3 (s-2)^2} \), the partial fractions expansion is \( \frac{A}{(s-1)} + \frac{B}{(s-1)^2} + \frac{C}{(s-1)^3} + \frac{D}{(s-2)} + \frac{E}{(s-2)^2} \).
While determining the constants \( A, B, C, D, \) and \( E \) requires solving a system of equations, understanding how to set up this system is crucial.
This method simplifies complex fractions into easier-to-handle parts, making them easier to integrate or work with in equations.