Problem 72 - Calculus Single Variable

Question

Computer algebra systems can calculate the partial fraction decomposition of a rational function when the arithmetic can be done with rational numbers. For example, the Maple command for obtaining the partial fraction decomposition of a rational expression \(R\) in the variable \(x\) is In Exercises \(72-75,\) use a computer algebra system to find the partial fraction decomposition of the given rational functions. $$ \frac{x^{3}-9 x^{2}+25 x-19}{x^{4}-5 x^{3}+6 x^{2}+4 x-8} $$

Step-by-Step Solution

Answer

Use a computer algebra system to find the partial fraction decomposition.

Step 1: Identify the Rational Function

Identify the given rational function, which is a fraction composed of a polynomial in the numerator and a polynomial in the denominator. The numerator is \(x^3 - 9x^2 + 25x - 19\) and the denominator is \(x^4 - 5x^3 + 6x^2 + 4x - 8\).

Step 2: Determine the Degree Conditions

Since the degree of the numerator (3) is less than the degree of the denominator (4), we can proceed directly to partial fraction decomposition without performing polynomial long division. In this step, we confirm that partial fraction decomposition is applicable without preliminary division.

Step 3: Use a Computer Algebra System (CAS)

Input the rational function into a computer algebra system. These systems are designed to perform symbolic computations, including partial fraction decomposition. For example, in Maple, you could use the command `convert((x^3-9*x^2+25*x-19)/(x^4-5*x^3+6*x^2+4*x-8), parfrac, x)`. This command directs the system to convert the rational function into its partial fraction form.

Step 4: Interpret the Output

After performing the computation in the CAS, you will receive the partial fraction decomposition of the given rational function. Examine the output provided by the CAS to understand the components of the decomposition. The result should be expressed as a sum of simpler fractions, each with denominator factors of the original function.

Key Concepts

Rational Functions

A rational function is a fraction where both the numerator and the denominator are polynomials. These functions are expressed in the form \( \frac{P(x)}{Q(x)} \), where \(P(x)\) and \(Q(x)\) are polynomials with \(Q(x) eq 0\).
In the given exercise, the rational function is \( \frac{x^{3}-9 x^{2}+25 x-19}{x^{4}-5 x^{3}+6 x^{2}+4 x-8} \), where the numerator is a cubic polynomial and the denominator is quartic.
Understanding the structure of a rational function is crucial, as it sets the foundation for various operations, such as finding asymptotes, intercepts, and performing partial fraction decomposition.

• Numerator: Polynomial \(P(x)\) in the expression.
• Denominator: Polynomial \(Q(x)\), crucial for division and determining undefined values.

Grasping these fundamental characteristics helps in manipulating and simplifying complex algebraic expressions.

Computer Algebra System

A computer algebra system (CAS) is a software tool used to perform symbolic computations. These systems are capable of handling algebraic expressions, performing complex calculations, and providing solutions to mathematical problems without manual arithmetic. Examples include MATLAB, Maple, and Mathematica.
The primary function of a CAS in this context is to simplify the process of partial fraction decomposition. By inputting a rational function into a CAS, one can quickly obtain its partial fraction decomposition.

• Allows for efficient solving of algebraic equations.
• Reduces potential errors often present in manual calculations.
• Supports educational purposes by providing step-by-step insights and solutions.

The ease of use and power of computer algebra systems make them indispensable tools for students and professionals in mathematics.

Polynomial Degree

The degree of a polynomial is the highest power of its variable; it gives vital information about the polynomial's properties. For instance, in the rational function \( \frac{x^{3}-9x^{2}+25x-19}{x^{4}-5x^{3}+6x^{2}+4x-8} \), the degree of the numerator is 3 and the degree of the denominator is 4.
These degrees play a significant role in determining whether partial fraction decomposition applies directly or requires manipulation beforehand, such as polynomial division.

• Numerator degree: Affects the setup of decomposition and initial evaluation premise.
• Denominator degree: Dictates the complexity and number of possible decomposed elements.

Recognizing polynomial degree impacts both the approach to problems and the interpretation of solutions, integrating seamlessly with the study of rational functions.

Symbolic Computation

Symbolic computation involves performing mathematical computations symbolically rather than numerically. This form of computation retains mathematical expressions in exact form, allowing for manipulation and algebraic simplification.
Tools like CAS leverage symbolic computation to execute tasks like partial fraction decomposition efficiently. Unlike numerical computation, symbolic methods maintain the integrity of exact values throughout operations.

• Advantages include precision and insight into underlying mathematical structures.
• Often used in algebra, calculus, and advanced mathematical programming.

Understanding symbolic computation is vital for interpreting and leveraging the full capabilities of modern mathematical software, especially when dealing with complex expressions.