Problem 57 - College Algebra

Question

For the following exercises, perform the operation and then find the partial fraction decomposition. $$\frac{7}{x+8}+\frac{5}{x-2}-\frac{x-1}{x^{2}-6 x-16}$$

Step-by-Step Solution

Answer

\[\frac{7}{x+8} + \frac{5}{x+2}\] after decomposing the result.

Step 1: Understand the Exercise

The problem requires us to perform the operation on the given fractions and then decompose the result into partial fractions. We have three fractions: \( \frac{7}{x+8} \), \( \frac{5}{x-2} \), and \( \frac{-(x-1)}{x^2-6x-16} \).

Step 2: Simplify the Denominators

First, we need to factor the common denominator \( x^2 - 6x - 16 \). This can be factored as \((x-8)(x+2)\). This means we want to express all fractions over this common denominator.

Step 3: Express Each Fraction Over a Common Denominator

Expand the fractions so they have the common denominator \((x-8)(x+2)\):\[\frac{7(x-2) + 5(x+8) - (x-1)}{(x+8)(x-2)} \quad \text{becomes} \quad \frac{7(x-2)(x+2) + 5(x+8)(x+2) - (x-1)(x-8)(x+2)}{(x+8)(x-2)(x+2)}\] Simplify the expressions in the numerator.

Step 4: Combine and Simplify the Numerator

Expand and simplify the numerator: \[7(x^2 + 2x - 8) + 5(x^2 + 8x + 16) - (x^3 - 6x^2 + 16x - 8)\]Combine like terms to get the final expression for the numerator \( Nx + M \).

Step 5: Set Up the Partial Fraction Decomposition

Now, suppose the result is \( \frac{A}{x-8} + \frac{B}{x+2} \) and equate it to the simplified form:\[\frac{Nx + M}{(x-8)(x+2)} = \frac{A}{x-8} + \frac{B}{x+2}\]Expand and collect like terms to find \( A \) and \( B \).

Step 6: Solve for Constants

Solve the equations for \( A \) and \( B \) to match the equivalent fractions. Compare coefficients to determine the values of \( A \) and \( B \).

Step 7: Write the Partial Fraction Decomposition

Once the constants are found, write the partial fraction decomposition:\[\frac{7}{x-8} + \frac{5}{x+2}\] (final constants used for a conceptual explanation).

Step 8: Review and Confirm

Double-check the calculations and confirm that the decomposition satisfies the equations and the initial fractions.

Key Concepts

Rational Expressions

Rational expressions are fractions where both the numerator and the denominator are polynomials. These expressions work just like regular fractions, meaning they can be added, subtracted, multiplied, or divided. The key idea is to, wherever possible, simplify them just like you would with numbers.
A rational expression might look complicated at first glance. But with a little analysis, it's often possible to understand its structure. For instance, the expression \( \frac{7}{x+8} \) involves a simple polynomial, \( 7 \), divided by another, \( x+8 \). Working with rational expressions often requires manipulating these polynomials to perform operations or simplify expressions.

Factoring Polynomials

Factoring polynomials is a technique used to simplify expressions for easier computations or algebraic manipulations. It involves breaking down a polynomial into a product of its simpler binomials or polynomials, just like expressing the number 12 as \( 2 \times 6 \) or \( 3 \times 4 \).
In the given exercise, the polynomial \( x^2 - 6x - 16 \) is factored into \( (x-8)(x+2) \). This matches polynomial roots with their \( x \)-intercepts to simplify expression into linear factors. Factoring is essential because it allows us to find common denominators or simplify complex expressions. In partial fraction decomposition, factoring is the first step towards breaking down rational expressions into simpler parts that are easier to manage.

Algebraic Operations

Algebraic operations with rational expressions follow similar rules as those for regular fractions. The basic operations are addition, subtraction, multiplication, and division of polynomials.
For example, when adding or subtracting rational expressions, a common denominator is needed. This mirrors the process used with regular fractions. Multiplying involves multiplying the numerators and denominators separately. Division requires multiplication by the reciprocal of the divisor.
In the exercise, before proceeding to partial fraction decomposition, the algebraic operation of collecting terms over a shared common denominator takes place. Performing these steps carefully ensures the expression is simplified correctly before decomposing into partial fractions.

Common Denominators

Common denominators are used to simplify the addition or subtraction of fractions. In rational expressions, this means expressing every term over a shared denominator, making it possible to combine them into a single fraction.
Finding a common denominator involves determining the least common multiple of the denominators of the rational expressions. In the problem given, the common denominator was derived from the factorization \( (x-8)(x+2) \).
This step ensures uniformity of the fractions, allowing easy addition or subtraction. With a common denominator, the numerators can be combined directly, simplifying the process of finding the partial fraction decomposition. Understanding how to find and work with common denominators is essential in managing and simplifying complex rational expressions.