Problem 30 - Precalculus: Mathematics for Calculus

Question

Find the partial fraction decomposition of the rational function. $$\frac{x-4}{(2 x-5)^{2}}$$

Step-by-Step Solution

Answer

The partial fraction decomposition is \(\frac{1}{2(2x-5)} + \frac{1/2}{(2x-5)^2}\).

Step 1: Identify the Denominator

The denominator of the rational function is \((2x - 5)^2\). This is a repeated linear factor.

Step 2: Set Up the Partial Fraction Format

Since the denominator is a repeated linear factor, the partial fraction format for the decomposition is \(\frac{x-4}{(2x-5)^2} = \frac{A}{2x-5} + \frac{B}{(2x-5)^2}\), where \(A\) and \(B\) are constants to be determined.

Step 3: Clear the Denominator

Multiply both sides by \((2x-5)^2\) to eliminate the denominators. This gives \(x-4 = A(2x-5) + B\).

Step 4: Expand and Combine Terms

Expand \(A(2x-5)\) to get \(2Ax - 5A\). Then, the equation \(x - 4 = 2Ax - 5A + B\) can be set up for comparison.

Step 5: Equate the Coefficients

Equate the coefficients of the powers of \(x\) on both sides. The equation becomes: \(2A = 1\) and \(-5A + B = -4\).

Step 6: Solve for A and B

From \(2A = 1\), solve for \(A\) to get \(A = \frac{1}{2}\). Substitute \(A\) into \(-5A + B = -4\) to find \(B = \frac{1}{2}\).

Step 7: Write the Partial Fraction Decomposition

Substitute \(A\) and \(B\) into the partial fraction format. The decomposition is \(\frac{x-4}{(2x-5)^2} = \frac{1}{2(2x-5)} + \frac{1/2}{(2x-5)^2}\).

Key Concepts

Rational Functions

Rational functions are expressions made of a ratio between two polynomials. These functions are represented as \( \frac{P(x)}{Q(x)} \), where both \(P(x)\) and \(Q(x)\) are polynomial expressions. The degree of the polynomial in the numerator (\(P(x)\)) is often equal to or less than the degree of the polynomial in the denominator (\(Q(x)\)).

Rational functions are essential because they appear in many real-world applications, from physics to finance. They can model rates of change and describe curves formed by ratios. Understanding rational functions helps in analyzing behavior, like asymptotes or points where the function is undefined due to division by zero. This makes being able to manipulate these expressions, such as through partial fraction decomposition, a key skill in calculus and algebra.

Repeated Linear Factors

In the decomposition of rational functions, repeated linear factors refer to terms in the denominator that occur more than once. For example, in the expression \((2x-5)^2\), the linear factor \(2x-5\) is repeated.

When dealing with repeated linear factors, the partial fraction decomposition involves setting up separate terms for each repetition of the linear factor. Thus, to decompose the function \( \frac{x-4}{(2x-5)^2} \), the decomposition would include \( \frac{A}{2x-5} + \frac{B}{(2x-5)^2} \), where each term corresponds to a repeated appearance of the factor in the denominator.

Understanding repeated linear factors is crucial because they require you to handle each power of the factor separately, ensuring you account for all terms in the rational function's expansion.

Coefficients Equating

Equating coefficients is a mathematical method used to find unknown values systematically. Once you have your partial fraction form, like \(A(2x-5) + B\) in this example, you need to equate the coefficients of corresponding powers of \(x\) on both sides of the equation.

• This involves:
• Expanding terms to align every part with its corresponding power of \(x\).
• Matching the coefficients of each power of \(x\) in the expanded equation to those in the original equation.

For example, you would equate \(2A\) to the coefficient of \(x\) (which is 1 in \(x-4\)), and solve for \(A\). Similarly, compare the constant terms to solve for \(B\).

This technique is essential in partial fraction decomposition and helps solve algebraic equations efficiently, enabling simplification and further manipulations.

Algebraic Fractions

Algebraic fractions refer to fractions where the numerator and/or the denominator are algebraic expressions, typically involving variables. In our context, these expressions are sometimes decomposed into simpler fractions — known as partial fractions — for ease of integration or solving limits.
Consider the fractional expression given by \( \frac{x-4}{(2x-5)^2} \). This is an algebraic fraction since both the numerator and the denominator contain algebraic expressions involving \(x\).
Rewriting algebraic fractions using partial fraction decomposition allows for easier manipulation, especially in calculus where solving for areas under curves or finding antiderivatives might be involved. It breaks down complex expressions into simpler, more manageable pieces, providing meaningful representations and aiding in the simplification of calculations.