Problem 4 - Precalculus: Functions and Graphs

Question

Find the partial fraction decomposition. \(\frac{5 x-12}{x^{2}-4 x}\)

Step-by-Step Solution

Answer

The partial fraction decomposition is \(\frac{3}{x} + \frac{2}{x-4}\).

Step 1: Factor the Denominator

First, we need to factor the denominator. The given denominator is \(x^2 - 4x\). This can be factored as \(x(x - 4)\).

Step 2: Set Up Partial Fractions

The goal is to express the fraction \(\frac{5x - 12}{x(x - 4)}\) as a sum of two simpler fractions. Since the degree of \(x(x - 4)\) is 2, we need two fractions: \(\frac{A}{x} + \frac{B}{x-4}\).

Step 3: Combine into a Single Fraction

Combine \(\frac{A}{x} + \frac{B}{x-4}\) into a single fraction using a common denominator: \(\frac{A(x-4) + Bx}{x(x-4)}\).

Step 4: Equate Numerators

Set the numerator of the original fraction equal to the new numerator: \(A(x-4) + Bx = 5x - 12\).

Step 5: Simplify and Collect Like Terms

Expand and collect like terms: \(Ax - 4A + Bx = 5x - 12\), which simplifies to \((A + B)x - 4A = 5x - 12\).

Step 6: Solve for Coefficients

From the equation \((A + B)x - 4A = 5x - 12\), we equate coefficients: - For \(x\) terms: \(A + B = 5\) - For constant terms: \(-4A = -12\) which gives \(A = 3\).

Step 7: Substitute and Solve for B

Substitute \(A = 3\) into \(A + B = 5\) to find \(B\): \(3 + B = 5\), which gives \(B = 2\).

Step 8: Write the Final Decomposition

Substitute \(A = 3\) and \(B = 2\) into the partial fractions: \(\frac{5x - 12}{x(x - 4)} = \frac{3}{x} + \frac{2}{x-4}\).

Key Concepts

Rational Expressions

Rational expressions are fractions that have polynomials in both the numerator and denominator. The expressions can represent real-world situations where certain variables are unknown. Working with rational expressions often requires applying algebraic techniques to simplify or solve equations.
In the given exercise, we deal with the rational expression \( \frac{5x - 12}{x^2 - 4x} \). Understanding how to manipulate such expressions is crucial in algebra, and partial fraction decomposition is a method to break these complex fractions into simpler parts. This technique is especially valuable when dealing with calculus and integrations.

Factoring Polynomials

Factoring is a method used to express a polynomial as a product of its simplest polynomials. This process is vital because it simplifies rational expressions, making them easier to handle.
In our example, we start by factoring the denominator \( x^2 - 4x \). Factoring yields \( x(x - 4) \), where \( x \) and \( x - 4 \) are simpler polynomials.

• Factoring reduces complex expressions into manageable pieces.
• For partial fraction decomposition, correct factoring is the first step toward simplifying the expression.

By factoring, we lay the groundwork for decomposing the rational expression into smaller, more manageable algebraic fractions.

Algebraic Fractions

Algebraic fractions are similar to regular fractions, but they contain variables as part of their expressions. The primary goal when working with these is to simplify or solve them.
In this exercise, we express a more complex fraction \( \frac{5x - 12}{x(x - 4)} \) as a sum of simpler algebraic fractions, \( \frac{3}{x} + \frac{2}{x-4} \). This simplification through partial fraction decomposition allows us to harness fundamental algebraic principles to solve equations more effectively. Understanding algebraic fractions and their manipulation is key to mastering rational expressions and making further mathematical operations more straightforward.

Numerator and Denominator

In any fraction, the numerator is the top part, and the denominator is the bottom part. When dealing with rational expressions, it's essential to manipulate both parts correctly to achieve the desired result.
For the given rational expression \( \frac{5x - 12}{x^2 - 4x} \), we focus on transforming the numerator \( 5x - 12 \) and the newly factored denominator \( x(x - 4) \). By rewriting the expression in the form \( \frac{3}{x} + \frac{2}{x-4} \), understanding how numerators and denominators interact becomes clearer.
Always remember:

• The degree of the numerator should be less than the degree of the denominator when decomposing fractions.
• Equating numerators is crucial for finding the coefficients needed for partial fraction decomposition.

Equation Solving

Equation solving is about finding values that satisfy a given equation. This process is often essential when working through partial fraction decomposition.
In this case, we equate the numerators from \( A(x-4) + Bx = 5x - 12 \). This step involves simplifying and comparing coefficients from both sides of the equation.
In our exercise, the solution involves:

• Equating the coefficients of \( x \): \( A + B = 5 \)
• Equating constant terms: \( -4A = -12 \), which gives \( A = 3 \)
• Substituting \( A \) to find \( B \): \( 3 + B = 5 \rightarrow B = 2 \)

By methodically solving these equations, we find the coefficients \( A \) and \( B \) that allow for the decomposition of the original fraction.