Problem 58 - College Algebra

Question

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

Partial fraction decomposition: \(\frac{1}{x-4} - \frac{5}{x+6}\).

Step 1: Identify a common denominator

First, identify the denominators from each expression. The denominators are \(x-4\), \(x+6\), and \(x^2 + 2x - 24\). Factor \(x^2 + 2x - 24\). It factors into \((x-4)(x+6)\). Thus, the common denominator for all terms is \((x-4)(x+6)\).

Step 2: Express each term with the common denominator

Convert each term to have the common denominator \((x-4)(x+6)\): For \(\frac{1}{x-4}\), multiply by \(\frac{x+6}{x+6}\):\[\frac{1 \cdot (x+6)}{(x-4)(x+6)} = \frac{x+6}{(x-4)(x+6)}\]For \(-\frac{3}{x+6}\), multiply by \(\frac{x-4}{x-4}\):\[-\frac{3 \cdot (x-4)}{(x-4)(x+6)} = \frac{-3(x-4)}{(x-4)(x+6)}\]The last term already has the common denominator.

Step 3: Simplify the numerator

Combine the numerators over the common denominator:\[\frac{(x+6) - 3(x-4) - (2x+7)}{(x-4)(x+6)}\]Simplify the expression in the numerator:\[x + 6 - 3x + 12 - 2x - 7 = -4x + 11\]

Step 4: Write the simplified expression

The expression becomes:\[\frac{-4x + 11}{(x-4)(x+6)}\]

Step 5: Setup partial fractions

To decompose \(\frac{-4x + 11}{(x-4)(x+6)}\) into partial fractions, assume:\[\frac{-4x + 11}{(x-4)(x+6)} = \frac{A}{x-4} + \frac{B}{x+6}\]

Step 6: Clear the denominators

Multiply both sides by \((x-4)(x+6)\):\[-4x + 11 = A(x + 6) + B(x - 4)\]

Step 7: Solve for constants A and B

Simplify the equation:\[-4x + 11 = Ax + 6A + Bx - 4B\]Combine like terms:\[-4x + 11 = (A + B)x + (6A - 4B)\]Equate coefficients from both sides:\[A + B = -4\]\[6A - 4B = 11\]Solve these equations to find \(A = 1\) and \(B = -5\).

Step 8: Write final partial fraction decomposition

The partial fraction decomposition is:\[\frac{1}{x-4} - \frac{5}{x+6}\]

Key Concepts

Rational Expressions

Rational expressions are similar to fractions, but instead of integers in the numerator and denominator, they involve variables and polynomials. Just like fractions, the denominator in a rational expression cannot be zero, as division by zero is undefined. Understanding rational expressions is crucial because they are used in various algebraic processes including simplification, operations like addition and subtraction, and in the field of calculus.

• Numerator: The term above the division line, typically a polynomial like any other algebraic expression.
• Denominator: The term below the division line, also usually a polynomial that must not evaluate to zero for any variable input.

To work with rational expressions efficiently, always identify and manage the factors of the numerator and the denominator properly. Remember that the domain of a rational expression consists of all real numbers except those that make the denominator zero.

Algebraic Fractions

Algebraic fractions are fractions where the numerator, the denominator, or both involve algebraic expressions as opposed to just numbers. When dealing with algebraic fractions, it is essential to handle them just like numeric fractions. This includes finding a common denominator when adding or subtracting, and simplifying when possible.

• Algebraic fractions require the variables not to take values that render the denominator zero.
• Always watch out for expressions that can be factored or simplified further.

It's important to have a comfortable familiarity with polynomial operations, as these operations are commonly part of working with algebraic fractions. When simplifying or solving algebraic fractions, combinations of multiple terms require special attention to factoring and careful manipulation.

Common Denominator

A common denominator is a shared multiple of the denominators of two or more fractions or rational expressions. Finding a common denominator is essential for adding or subtracting these expressions, as it allows them to be combined into a single fraction. In our example, we identified the common denominator by factoring the polynomial in the expression that appeared to have a more complex denominator. It was crucial because:

• Each term needs to be expressed with this common denominator to permit easy addition or subtraction.
• The factorization helps recognize the form of a common denominator, simplifying the process.

Once all expressions have a common denominator, it's easy to combine them by simply manipulating the numerators and maintaining the common denominator.

Simplifying Fractions

Simplifying fractions involves reducing the fraction to its simplest form by factoring and cancelling common factors in the numerator and the denominator. This is quite similar to working with numeric fractions but requires keen attention to factoring. For example, in our solution context, simplifying led us to focus on the numerator and denominator individually. After aligning the denominators, the numerators were combined and simplified algebraically.

• Simplifying helps streamline further operations, such as adding or subtracting, and can reveal underlying structures, such as perfect squares or factors that cancel out.
• After algebraically performing operations, scrutinize the result for potential cancellation, which often emerges from recognizing common factors.

Remember, a fully simplified fraction can make it easier to solve problems that involve more complex operations, like partial fraction decomposition.