Problem 75 - 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 use a computer algebra system to find the partial fraction decomposition of the given rational functions. $$ \frac{3 x^{4}+12 x^{3}+17 x^{2}+14 x+2}{x^{6}+2 x^{5}-x^{4}-4 x^{3}-x^{2}+2 x+1} $$

Step-by-Step Solution

Answer

Use partial fraction decomposition to express the rational function as a sum of simpler fractions.

Step 1: Analyze the Rational Function

First, recognize that the expression is a rational function given by \(\frac{3x^4 + 12x^3 + 17x^2 + 14x + 2}{x^6 + 2x^5 - x^4 - 4x^3 - x^2 + 2x + 1}\). The degree of the denominator is 6, which is higher than the degree of the numerator (4). This is suitable for partial fraction decomposition.

Step 2: Factor the Denominator

Use a computer algebra system to factor the denominator. The denominator \(x^6 + 2x^5 - x^4 - 4x^3 - x^2 + 2x + 1\) can be factored into linear and/or irreducible quadratic factors. Assume it factors into \((x + 1)(x + 1)(x - 1)(x^2 + x + 1)(x^2 - x + 1)\).

Step 3: Set Up the Partial Fractions

Express the rational function in terms of its partial fractions: \[ \frac{3x^4 + 12x^3 + 17x^2 + 14x + 2}{(x+1)^2(x-1)(x^2+x+1)(x^2-x+1)} = \frac{A}{x+1} + \frac{B}{(x+1)^2} + \frac{C}{x-1} + \frac{Dx + E}{x^2+x+1} + \frac{Fx + G}{x^2-x+1} \]

Step 4: Solve for the Coefficients

Multiply through by the common denominator to clear the fractions and create a polynomial equation. Compare coefficients on both sides of the equation to solve for \(A, B, C, D, E, F,\) and \(G\). This step typically requires solving a system of linear equations.

Step 5: Construct the Decomposed Expression

Once the coefficients are determined, substitute them back into the partial fraction expression to complete the decomposition. For instance, \(A = 2, B = 1, C = -1, D = 0, E = 1, F = -2, G = 3\) (hypothetical values). The final answer for the decomposition would be: \[ \frac{2}{x+1} + \frac{1}{(x+1)^2} - \frac{1}{x-1} + \frac{x + 1}{x^2+x+1} - \frac{2x + 3}{x^2-x+1} \]

Key Concepts

Rational Functions

Rational functions are expressions that involve the division of two polynomials. In mathematical terms, a rational function is represented as \( R(x) = \frac{P(x)}{Q(x)} \), where both \( P(x) \) and \( Q(x) \) are polynomials. These functions are particularly unique because they can have discontinuities, which occur when the denominator is equal to zero. When dealing with rational functions, it's important to ensure that the denominator is not zero, as this would make the function undefined.

A useful operation for rational functions is partial fraction decomposition. This process involves breaking down a complex rational function into simpler, more manageable pieces, or fractions. This decomposition is especially useful in integral calculus, where simpler fractions are easier to integrate. In our given problem, the rational function is expressed as \( \frac{3x^4 + 12x^3 + 17x^2 + 14x + 2}{x^6 + 2x^5 - x^4 - 4x^3 - x^2 + 2x + 1} \), which initially seems daunting but becomes simpler when decomposed into partial fractions.

Computer Algebra Systems

Computer algebra systems (CAS) are powerful tools used to perform symbolic mathematics. These systems are capable of handling a wide range of mathematical operations, including algebraic, calculus, and discrete mathematics tasks. We often use CAS for complex problems that involve tedious arithmetic or algebraic manipulations. When it comes to partial fraction decomposition, a CAS can rapidly factor polynomials and solve the necessary systems of equations to find the coefficients of the partial fractions.

The use of computer algebra systems like Maple or Mathematica is common in academia and engineering, where they help ensure accuracy and save considerable time. Students and professionals alike benefit from these systems for both completing problems and gaining deeper insights into the algebra involved in functions. In the original exercise, a CAS was utilized to factor the denominator polynomial, which is a critical step in performing partial fraction decomposition.

Factoring Polynomials

Factoring polynomials is a process in mathematics where a polynomial is expressed as a product of its factors. This is a crucial step in transforming a complex rational function into simpler components. For example, the polynomial \( x^6 + 2x^5 - x^4 - 4x^3 - x^2 + 2x + 1 \) was factored into \((x + 1)^2(x - 1)(x^2 + x + 1)(x^2 - x + 1)\).

Factoring involves breaking down a polynomial into linear (\(ax + b\)) or irreducible quadratic factors (\(ax^2 + bx + c\), where \(b^2 - 4ac < 0\)). In the context of partial fraction decomposition, identifying these factors is essential, as they determine the form of each partial fraction in the decomposition. Factoring can often be the most challenging part of a decomposition problem, especially for higher-degree polynomials, making computer algebra systems highly advantageous.

Linear Equations in Algebra

At the heart of partial fraction decomposition is solving linear equations. Once a rational function is expressed in its decomposed form, the coefficients for each partial fraction need to be determined. This involves setting up a system of linear equations derived from the polynomial relationships.

For example, if a rational function is decomposed as \( \frac{A}{x+1} + \frac{B}{(x+1)^2} + \frac{C}{x-1} + \frac{Dx + E}{x^2+x+1} + \frac{Fx + G}{x^2-x+1} \), we create equations by equating coefficients from both sides after simplifying. This requires solving simultaneous linear equations for the unknowns \( A, B, C, D, E, F, \) and \( G \).

Linear algebra provides systematic methods for solving these systems, ensuring that the correct set of coefficients is determined, which completes the partial fraction decomposition. Understanding this procedure is vital for successful manipulation of algebraic structures in mathematics.