Problem 83 - Introductory Algebra for College Students

Question

Perform the indicated operation or operations. Simplify the result, if possible. $$\frac{x+6}{x^{2}-4}-\frac{x+3}{x+2}+\frac{x-3}{x-2}$$

Step-by-Step Solution

Answer

The simplified form of the expression is \( 6x / (x+2)(x-2) \).

Step 1: Factorize the denominators and find the least common denominator (LCD)

We first notice that the denominator \(x^{2}-4\) of the first fraction can be factorized into (x+2)(x-2). So, the least common denominator is (x+2)(x-2). In the second and third fractions, the denominator can be considered as such.

Step 2: Express each fraction with the least common denominator (LCD)

The first fraction already has the LCD. Rewrite the second fraction as \((x+3)(x-2)/(x+2)(x-2)\) and the third fraction as \((x-3)(x+2)/(x+2)(x-2)\) respectively. Now, all fractions have the common denominator (x+2)(x-2).

Step 3: Combine and simplify

Combine the three fractions and simplify the numerator by combining like terms, resulting in \( (x+6 - (x+3)(x-2) + (x-3)(x+2)) / (x+2)(x-2) \). This simplifies to \( (x+6 - x^{2}+5x-6 + x^{2}-x-6) / (x+2)(x-2) \) which further simplifies to \( 6x / (x+2)(x-2) \).

Key Concepts

Factoring Polynomials

Factoring polynomials is like finding the building blocks of a polynomial expression. This method involves breaking down a polynomial into several simpler polynomials that, when multiplied together, produce the original polynomial. In our exercise, this concept plays a crucial role. For instance, the term \( x^2 - 4 \) was factored into \((x+2)(x-2)\). This technique uses the difference of squares identity, which states that \( a^2 - b^2 = (a-b)(a+b) \). It's important to recognize and use factoring patterns like these to quickly simplify expressions.

• Look for patterns such as difference of squares, perfect square trinomials, or common factors.
• Factoring is especially helpful when simplifying complex rational expressions.

Understanding these patterns makes simplifying expressions and finding common denominators easier, aiding in operations like subtraction and addition of fractions.

Least Common Denominator

The least common denominator (LCD) is pivotal when dealing with the addition or subtraction of fractions because it allows us to combine them. In algebraic fractions, the LCD is the smallest expression that each denominator can divide into without a remainder. It requires identifying the common factors from the factored forms of the denominators.
In this exercise, we identified the common denominator as \((x+2)(x-2)\) after factoring \(x^2-4\). This means that each fraction involved in the operation must be expressed with this common denominator.

• To find the LCD, first factor all denominators completely.
• Combine all unique factors using the highest power each appears in any denominator.
• Rewrite each fraction so that it has the LCD.

This process is essential for aligning fractions under a common denominator, thereby allowing them to be added or subtracted effectively.

Simplifying Algebraic Fractions

Simplifying algebraic fractions involves reducing them to their simplest form by canceling out common terms. Once all fractions have a common denominator, their numerators can be combined, and simplification is possible. In the given problem, fractions were combined and simplified to produce a single fraction with the least denominator.

• Combine the numerators by performing the indicated operations (addition or subtraction).
• Look out for common terms in the new polynomial numerator that can be canceled with terms in the denominator.
• The result should be a simplified fraction, showing the simplest representation of the solution.

By simplifying, we ensure the expression is as concise as possible, making analysis and further operations simpler and more intuitive.