Problem 24 - Precalculus

Question

(a) Find an appropriate viewing rectangle to demonstrate that the following purported partial fraction decomposition is incorrect: $$\frac{4}{x^{2}(x-5)}=\frac{-4 / 5}{x^{2}}+\frac{4 / 25}{x-5}$$ (b) Follow part (a) using $$\frac{4}{x^{2}(x-5)}=\frac{-3 / 25}{x}+\frac{-2 / 5}{x^{2}}+\frac{6 / 25}{x-5}$$ (c) Determine the correct partial fraction decomposition. given that it has the general form $$\frac{4}{x^{2}(x-5)}=\frac{A}{x}+\frac{B}{x^{2}}+\frac{C}{x-5}$$

Step-by-Step Solution

Answer

Both given decompositions are incorrect. The correct decomposition is \(\frac{-2/5}{x} + \frac{-4/5}{x^2} + \frac{2/5}{x-5}\).

Step 1: Understanding the Initial Problem

We need to check the correctness of the given partial fraction decompositions. Begin by examining part (a). The expression is supposed to be:\[\frac{4}{x^{2}(x-5)}=\frac{-4/5}{x^{2}}+\frac{4/25}{x-5}\]This implies that if we equate this to the original function and simplify, the result should still be \(\frac{4}{x^2(x-5)}\).

Step 2: Simplify the Given Decomposition in Part (a)

Let's clear the denominators by multiplying through by \(x^2(x-5)\):\[\frac{-4}{5} \cdot (x-5) + \frac{4}{25} \cdot x^2\]Simplifying this gives:\(-\frac{4x}{5} + 4 + \frac{4}{25}x^2\)Now, compare this with \(\frac{4}{x^2(x-5)}\) and verify if these two expressions are equivalent. They are not, because the simplified expression does not reconstruct the original function, indicating an error.

Step 3: Evaluate the Decomposition in Part (b)

The new equation is:\[\frac{4}{x^{2}(x-5)}=\frac{-3/25}{x}+\frac{-2/5}{x^{2}}+\frac{6/25}{x-5}\]Clear the fractions by multiplying by \(x^2(x-5)\) to get:\[\frac{-3}{25}x (x-5) + \frac{-2}{5}(x-5) + \frac{6}{25}x^2\]Simplifying this gives:\(-\frac{3}{25}x^2 + \frac{3}{5}x + \frac{12}{5}\)Compare this with \(\frac{4}{x^2(x-5)}\). Again they are not equal, which shows this decomposition is also incorrect.

Step 4: Find the Correct Partial Fraction Decomposition

Assume:\[\frac{4}{x^{2}(x-5)}=\frac{A}{x}+\frac{B}{x^{2}}+\frac{C}{x-5}\]By clearing the denominators, multiply through by \(x^2(x-5)\) to get:\[Ax(x-5) + B(x-5) + Cx^2 = 4\]This expression equals:\[Ax^2 - 5Ax + Bx - 5B + Cx^2\]Combine like terms and equate coefficients of corresponding powers of \(x\) to solve for \(A, B, C\). This results in a system of equations:1. \(A + C = 0\)2. \(-5A + B = 0\)3. \(-5B = 4\)Solving these equations, we find that \(A = -\frac{2}{5}, B = -\frac{4}{5}, C = \frac{2}{5}\).

Step 5: Verify the Correct Decomposition

Insert the values of \(A, B,\) and \(C\) back into the partial fraction form to ensure consistency:\[\frac{4}{x^{2}(x-5)} = \frac{-2/5}{x} + \frac{-4/5}{x^2} + \frac{2/5}{x-5}\]Combine these terms and simplify to verify it reconstructs the original expression.

Key Concepts

Rational Functions

A rational function is a fraction where both the numerator and the denominator are polynomials. Understanding rational functions is crucial when working with partial fraction decomposition, as it involves breaking down a complex rational function into simpler fractions, which are easier to handle.
For example, let's consider the function you might encounter:

• The numerator is 4, a constant polynomial.
• The denominator is the polynomial, consisting of factors like \(x^2\) and \(x - 5\).

This type of function requires careful analysis to break down accurately using partial fraction decomposition. It's especially important to confirm that any adjustments or simplifications made to the rational function correctly rebuild the original expression. Understanding the structure and behavior of rational functions sets the foundation for the following steps in decomposition.

Algebraic Manipulation

Algebraic manipulation is a powerful technique to simplify expressions. It involves rearranging and modifying algebraic expressions to reach a particular form.
Let's explore some key operations involved in the process:

• Clearing denominators: Multiply through by a factor to eliminate fractions, making it easier to work with polynomials.
• Combining like terms: Group similar terms to simplify expressions, aligning terms with common variables or constants.

In partial fraction decomposition, algebraic manipulation ensures that the complex rational expression simplifies to match the original function's form.
Practicing these manipulations helps in managing more complex algebraic tasks, such as solving equations and verifying solutions.

Polynomial Division

Polynomial division is used to divide a polynomial by another polynomial, helping to determine the remainder or factor of the division. In the context of partial fraction decomposition, polynomial division assists in simplifying the expression before breaking it into simpler fractions.
Consider a scenario where the numerator's degree is higher than or equal to the denominator's degree:

• In such cases, divide the numerator by the denominator to simplify the expression.
• This process may result in a remainder, which plays a crucial part in further simplification.

Understanding polynomial division allows you to reduce expression complexity, thereby easing the path toward efficient decomposition.
It's much like traditional numerical division, but it requires careful attention to variable powers in the polynomials.

System of Equations

When finding the corrections in partial fraction decomposition, leveraging a system of equations becomes essential. It often arises when you equate coefficients of a decomposed expression to those of the original function.
Here’s an overview of how the process works:

• Express the decomposed chunks in a structure like \(Ax(x-5) + B(x-5) + Cx^2 = 4\).
• Match and set up equations based on coefficients of powers of \(x\), capturing all terms.

This leads to creating equations for each power of \(x\), a straightforward way of solving for variables \(A, B, and C\).
Solving these linear equations provides the values required for the correct partial structure.Systems of equations in this context are pivotal, as they ensure each component adds up to recreate the full form of the original expression.