Problem 25 - College Algebra

Question

Find the decomposition of the partial fraction for the repeating linear factors. \(\frac{5-x}{(x-7)^{2}}\)

Step-by-Step Solution

Answer

Decomposition: \(\frac{-1}{x-7} + \frac{-2}{(x-7)^2}\).

Step 1: Identify the Partial Fraction Form

The given expression is \(\frac{5-x}{(x-7)^2}\). For a repeated linear factor \((x-a)^2\), the partial fraction decomposition will be \(\frac{A}{x-7} + \frac{B}{(x-7)^2}\).

Step 2: Write the Equation

Set up the equation \(\frac{5-x}{(x-7)^2} = \frac{A}{x-7} + \frac{B}{(x-7)^2}\). This states that the rational expression is equal to the sum of the two partial fractions.

Step 3: Clear the Denominator

Multiply both sides by \((x-7)^2\) to remove the denominator: \(5-x = A(x-7) + B\).

Step 4: Expand and Simplify

Expand the expression on the right-hand side: \(A(x-7) = Ax - 7A\). So we get \(5-x = Ax - 7A + B\).

Step 5: Equate and Solve for Coefficients

Compare coefficients: 1. For the \(x\) terms: \(-1 = A\)2. For the constant terms: \(5 = -7A + B\)Substitute \(A = -1\) into the second equation: \(5 = -7(-1) + B\). This gives \(5 = 7 + B\), so \(B = -2\).

Step 6: Write the Final Decomposition

The decomposition is \(\frac{-1}{x-7} + \frac{-2}{(x-7)^2}\).

Key Concepts

Repeated Linear Factors

Repeated linear factors occur when a polynomial denominator in a rational expression includes a factor that is raised to a power greater than one. This is represented as

• e.g., \((x-a)^n\) where \(n\) is greater than 1.
• This means the same factor, \((x-a)\), is multiplied by itself \(n\) times.

When dealing with repeated linear factors, partial fraction decomposition requires a unique setup. Each instance of the repeated factor will contribute a separate term to the decomposition, with varying powers of the factor in the denominator.For example, for a factor like \((x-7)^2\), the decomposition form is:

• \(\frac{A}{x-7}\) for the first power
• \(\frac{B}{(x-7)^2}\) for the second power

In our original problem, \((x-7)^2\) required terms \(\frac{A}{x-7}\) and \(\frac{B}{(x-7)^2}\) to represent the repeated factors appropriately.This method efficiently handles every level of repetition and ensures the rational expression can be decomposed into simpler parts.

Decomposition of Rational Expressions

Decomposition of rational expressions involves breaking down a complicated rational expression into a sum of simpler fractions.The goal is to rewrite the expression in a way that is easier to work with, especially for integration or solving equations.The expression is split into partial fractions, each with its own distinct denominator.In our exercise, \(\frac{5-x}{(x-7)^2}\), we are working with a repeated linear factor.Using partial fraction decomposition, we set:

• \(\frac{5-x}{(x-7)^2} = \frac{A}{x-7} + \frac{B}{(x-7)^2}\)

This method allows us to solve for constant coefficients, \(A\) and \(B\), by eliminating the denominators and comparing coefficients.The decomposition includes distinct fractions, each taking a part of the original expression:

• The breakdown into easier terms means specific coefficients \(A\) and \(B\) need to be calculated to ensure the decomposition equals the original expression.

This step-by-step calculation makes the expressions manageable and simpler for further operations, like integration, where dealing with smaller fractions is advantageous.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations (addition, subtraction, multiplication, and division).These expressions can represent all sorts of mathematical concepts and problems.In calculus, algebraic expressions are often manipulated into different forms to simplify calculation and analysis.The original expression \(\frac{5-x}{(x-7)^2}\)is an example where algebra is used to restructure for the purpose of simplification.Understanding algebraic expressions involves various operations:

• Identifying terms involving variables and their coefficients – like recognizing \(-x\) and its coefficient in the expression is \(-1\).
• Grouping terms methodically by their degree (e.g., linear, quadratic).
• Applying fundamental operations to simplify or alter the expressions for practical use in solving or integrating.

By mastering these skills, students can efficiently deal with any given rational expression and solve it by partial fraction decomposition or other algebraic methods.Ultimately, algebra forms the backbone that supports more advanced concepts in mathematics, making understanding and working with expressions essential for mathematical proficiency.