Problem 58 - College Algebra with Corequisite Support

Question

For the following exercises, perform the operation and then find the partial fraction decomposition. $$ \frac{1}{x-4}-\frac{3}{x+6}-\frac{2 x+7}{x^{2}+2 x-24} $$

Step-by-Step Solution

Answer

Perform partial fraction decomposition with results: \(\frac{5}{2(x-4)} - \frac{13}{2(x+6)}\).

Step 1: Simplify the Expression

Combine all the terms over a common denominator. The denominators in the expression are \((x-4)\), \((x+6)\), and \((x^2+2x-24)\). First, factor \(x^2 + 2x - 24\) into \((x-4)(x+6)\), which is the product of the other two denominators.

Step 2: Express Each Term with the Common Denominator

Re-write each fraction so that they all have the common denominator \((x-4)(x+6)\). The full expression will be: \[ \frac{1}{x-4} = \frac{x+6}{(x-4)(x+6)}, \quad \frac{3}{x+6} = \frac{3(x-4)}{(x-4)(x+6)}, \quad \frac{2x+7}{x^2 + 2x-24} = \frac{2x+7}{(x-4)(x+6)} \]

Step 3: Combine the Fractions

Add the fractions together: \[ \frac{x+6 - 3(x-4) - (2x+7)}{(x-4)(x+6)} \]Simplify the numerator:\[ x+6 - 3x + 12 - 2x - 7 = -4x + 11 \]So you have:\[ \frac{-4x + 11}{(x-4)(x+6)} \]

Step 4: Set Up the Partial Fraction Decomposition

Express \(\frac{-4x + 11}{(x-4)(x+6)}\) as a sum of partial fractions:\[ \frac{-4x + 11}{(x-4)(x+6)} = \frac{A}{x-4} + \frac{B}{x+6} \]

Step 5: Solve for A and B

Multiply through by the common denominator \((x-4)(x+6)\) to clear fractions:\[ -4x + 11 = A(x+6) + B(x-4) \]Expand and solve the equations:\[ -4x + 11 = Ax + 6A + Bx - 4B \]Combine like terms:\[ -4x + 11 = (A+B)x + (6A - 4B) \]Set up the system of equations:1. \(A + B = -4\)2. \(6A - 4B = 11\) Solve for \(A\) and \(B\), resulting in \(A = \frac{5}{2}\), \(B = -\frac{13}{2}\).

Step 6: Write the Partial Fractions

Substitute \(A\) and \(B\) back into the partial fraction decomposition:\[ \frac{-4x + 11}{(x-4)(x+6)} = \frac{\frac{5}{2}}{x-4} + \frac{-\frac{13}{2}}{x+6} \]

Key Concepts

Rational Expressions

A rational expression is essentially a fraction in which both the numerator and the denominator are polynomials. Understanding rational expressions is crucial for solving a multitude of algebraic problems, including partial fraction decomposition. When dealing with rational expressions, ensure that the denominator does not equal zero, as this would make the expression undefined.
When working with rational expressions, it helps to:

• Simplify the expression before performing any operations.
• Factor any polynomials in the numerator and the denominator to simplify further if possible.
• Find a common denominator when adding or subtracting rational expressions to combine them efficiently.

In the original exercise, the rational expression is composed of separate fractions that need to be combined over a common denominator before proceeding with partial fraction decomposition.

Common Denominator

To combine rational expressions, especially those with different denominators, finding a common denominator is necessary. A common denominator is a shared multiple that each of the original denominators can divide into evenly. This allows multiple expressions to be combined into a single fraction, simplifying further manipulations.
To find a common denominator:

• Factor each denominator fully.
• Identify all unique factors across the denominators.
• Use the highest power of each factor when constructing the common denominator.

In the exercise, the expression \( \frac{2x+7}{x^2 + 2x-24} \) provides the factored form \((x-4)(x+6)\), serving as the common denominator for all terms. This simplification is essential before combining the fractions into one numerator.

Factoring Quadratics

Factoring quadratics is a process used to simplify expressions, especially when seeking to identify a common denominator for rational expressions. Quadratic equations typically follow the form \(ax^2 + bx + c\). To factor them, you look for two numbers that multiply to \(ac\) and add to \(b\).
In the case of the quadratic polynomial \(x^2 + 2x - 24\) in the exercise, it is factored into \((x-4)(x+6)\). Here’s how you can factor such an expression:

• Identify \(a\), \(b\), and \(c\) (1, 2, and -24, respectively, in this case).
• Find two numbers whose product is \(a \cdot c = -24\) and whose sum is \(b = 2\).
• The numbers -4 and 6 satisfy these conditions. Therefore, the factored form is \((x-4)(x+6)\).

Factoring is a pivotal skill in algebra that simplifies complex expressions and aids in solving equations more efficiently.

Systems of Equations

Solving a system of equations is a necessary step in decomposing a fraction into partial fractions. This process is used to find the coefficients (constants) of the terms in the decomposition, allowing the original fraction to be expressed as a sum of simpler fractions.
The task involves:

• Setting up equations by equating the original fraction’s polynomial (numerator) to the sum of the numerators of the partial fractions multiplied by their respective missing denominators.
• In the exercise, -4x + 11 = Ax + 6A + Bx - 4B is expanded and grouped by terms.
• By comparing coefficients from both sides, we establish a system of equations: \(A + B = -4\) and \(6A - 4B = 11\).
• Solve the system to find values of the constants; here, \(A = \frac{5}{2}\) and \(B = -\frac{13}{2}\).

Effectively solving a system of equations is crucial for correctly expressing the original rational expression as partial fractions, simplifying further analysis or integration processes.