Problem 23 - College Algebra

Question

Write the partial fraction decomposition of each rational expression. $$\frac{x^{2}-6 x+3}{(x-2)^{3}}$$

Step-by-Step Solution

Answer

We can write the given rational expression \(\frac{x^{2}-6 x+3}{(x-2)^{3}}\) as \(1 - \frac{2}{x-2} - \frac{1}{(x-2)^{2}} - \frac{1}{(x-2)^{3}}\).

Step 1: Perform Long Division

Since the degree of the numerator is equal to the degree of the denominator, the first step is to perform long division. This results in \(1 - \frac{8x - 7}{(x-2)^{3}}\). Now, we only need to decompose \(\frac{8x - 7}{(x-2)^{3}}\).

Step 2: Decompose into Partial Fractions

We decompose \(\frac{8x - 7}{(x-2)^{3}}\) into partial fractions. Since the denominator of the fraction is a repeated root, we write each fraction with \(x-2\), \((x-2)^{2}\), and \((x-2)^{3}\) in the denominator and unknown constants in the numerators, giving us \(\frac{A}{x-2} + \(\frac{B}{(x-2)^{2}}\) + \(\frac{C}{(x-2)^{3}}\).

Step 3: Solving for the Coefficients

Next, to find the coefficients we multiply throughout by \((x-2)^{3}\) to clear the denominator, which results in \(8x - 7 = A(x-2)^{2} + B(x-2) + C\). Setting x = 2 provides \(C=1\). Differentiating both sides twice after simplification gives A and B as -2, -1 respectively.

Key Concepts

Long Division

Long division is a mathematical technique similar to division of numbers. Here, it's used for dividing polynomials. In the original exercise, we have a rational expression where the degree of the numerator is equal to that of the denominator. Thus, long division is necessary to simplify the expression.

Through long division, we divide the numerator by the denominator until the degree of the remainder is less than the degree of the denominator.

• This simplifies the process into smaller components that are easier to manage.
• In the given example, it results in the expression: \(1 - \frac{8x - 7}{(x-2)^{3}}\).

The leftover fraction after doing long division is what we work with further to perform partial fraction decomposition.

Repeated Roots

When dealing with polynomials, a repeated root is a root that occurs more than once. In our exercise, the factor \((x-2)\) in the denominator is raised to the third power. This indicates it is a repeated root.

Here's what that means for partial fraction decomposition:

• The repeated root necessitates writing separate terms for each power of the root.
• Each term's denominator will contain one, two, or all powers of the repeated root. For example, \( \frac{A}{x-2} + \frac{B}{(x-2)^{2}} + \frac{C}{(x-2)^{3}} \).

This setup allows clarity and completeness of the solution, ensuring all possible solutions for the coefficients cover the repeated root adequately.

Solving for Coefficients

Once we set up the partial fractions, the next step is to solve for the coefficients. The goal is to express the decomposed form in terms of the unknown coefficients \(A\), \(B\), and \(C\).

Here's how to find these coefficients:

• Multiply throughout by the common denominator to eliminate fractions. This gives an algebraic equation.
• Substitute convenient values for \(x\) to solve for each coefficient separately. Often, setting \(x\) equal to the root simplifies the process.
• In this exercise, by setting \(x = 2\), \(C\) can be found directly.
• Further calculations, such as differentiation, are used to efficiently determine \(A\) and \(B\).

Systematic calculation ensures that each coefficient fully represents the true components of the original rational expression.

Rational Expressions

Rational expressions are fractions where both the numerator and the denominator are polynomials. They form a crucial part of algebra and are often encountered in calculus.

In the exercise at hand, \(\frac{x^{2}-6x+3}{(x-2)^{3}}\) is a rational expression. To work with them effectively:

• Consider the degrees of the polynomials involved to decide on simplification methods.
• Long division helps when the numerator's degree is equal to or higher than the denominator's.
• Partial fraction decomposition breaks complex expressions into simpler fractions, especially useful for integration.

Understanding how to manage rational expressions is a vital skill, allowing more manageable analysis and problem-solving.