Question

Finding constants with a computer algebra system Give the -appropriate form of the partial fraction decomposition of the expression, and then use a computer algebra system to find the unknown constants. $$\frac{x^{4}+3 x^{2}+1}{x\left(x^{2}+1\right)^{2}\left(x^{2}+x+4\right)^{2}}$$

Short answer

Expression: $$\frac{x^{4}+3x^{2}+1}{x\left(x^{2}+1\right)^{2}\left(x^{2}+x+4\right)^{2}}$$ Answer: The partial fraction decomposition of the given expression is: $$\frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{(x^2 + 1)^2} + \frac{Fx + G}{x^2 + x + 4} + \frac{Hx + I}{(x^2 + x + 4)^2}$$ where A, B, C, D, E, F, G, H, and I are constants that can be found using a computer algebra system (CAS).

Step-by-step solution

Find the appropriate form of the partial fraction decomposition

We have the given expression: $$\frac{x^{4}+3x^{2}+1}{x\left(x^{2}+1\right)^{2}\left(x^{2}+x+4\right)^{2}}$$ We'll need to break the expression into partial fractions of the form: $$\frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{(x^2 + 1)^2} + \frac{Fx + G}{x^2 + x + 4} + \frac{Hx + I}{(x^2 + x + 4)^2}$$

Use the computer algebra system to find the constants

At this point, you will need to use a computer algebra system to find the values of the unknown constants A, B, C, D, E, F, G, H, and I. You can use software such as Mathematica, Maple, or an online CAS like Sympy. Simply input the following command to find the constants: `apart((x**4 + 3*x**2 + 1)/(x*(x**2 + 1)**2*(x**2 + x + 4)**2))` Note: The syntax may vary slightly depending on the computer algebra system you're using. Be sure to consult the documentation for the appropriate syntax.

Key concepts

Computer Algebra System

A Computer Algebra System (CAS) is a powerful tool used to perform symbolic mathematics by automating tedious algebraic manipulations. For the given problem, using a CAS helps in the decomposition of a rather complex rational expression into simpler fractions.
The main advantages of using a CAS include:

• Efficiency: It processes calculations much faster than manual computations.
• Accuracy: It reduces human error in complex calculations.
• Versatility: Many algebra systems support a wide range of mathematical operations, from basic arithmetic to advanced calculus and linear algebra.

Examples of popular CAS include Mathematica, Maple, and SymPy. By using one of these systems to input the desired expression, students can quickly see the breakdown of their equation into partial fractions and find unknown constants precisely. Simply enter a command, like the one shown in the exercise 'apart' function in SymPy, and let the software work its magic. This is especially helpful for validating manual calculations against the results provided by the CAS.

Mathematical Constants

Mathematical constants are specific numbers with fixed values, often appearing in various fields of mathematics. Although the concept comes into play more prominently in other branches like calculus, they are essential even in algebra, particularly for distinguishing between coefficients in partial fraction decomposition.
In the context of the given exercise, constants such as A, B, C, ..., I are part of the partial fraction decomposition. These alphabetical symbols represent unknown numerical values that must be determined to split the larger expression into its partial components. Once inputted into a CAS, these symbols will be replaced with actual numeric values—turning abstract symbols into real numbers, which can aid in solving and understanding equations further in mathematical analysis. Remember that these constants vary between decompositions, reflecting the unique structure and behavior of each individual rational function.

Mathematical Software

Mathematical software refers to specialized programs or tools designed to solve mathematical problems and perform various computations. These are crucial aids in classrooms and research environments, especially when dealing with intricate problems like partial fraction decomposition.
Here's why mathematical software is beneficial:

• Visualization: Some programs offer graphical representations, providing insights that numbers alone can't reveal.
• Accessibility: Many software tools, including open-source ones like SymPy, are available online, making powerful mathematical tools accessible to anyone.
• Comprehensive resources: Software often comes with detailed documentation and communities that offer support and educational resources.

Using mathematical software to solve problems not only improves efficiency and accuracy but also enriches learning by providing step-by-step solutions and explanations. These capabilities make mathematical software an indispensable resource for both students tackling homework exercises and professionals conducting research.