Problem 11 - Algebra 2

Question

Simplify each expression. $$ \frac{1}{x^{2}-4}+\frac{x}{x+2} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{x^2 - 2x + 1}{(x-2)(x+2)} \).

Step 1: Factor the Denominator

First, note that the denominator in the first fraction can be factored. Recall the difference of squares: \[ x^2 - 4 = (x - 2)(x + 2). \]So, rewrite the first fraction as: \[ \frac{1}{(x-2)(x+2)}. \]

Step 2: Identify the Least Common Denominator (LCD)

The LCD of the expressions \( \frac{1}{(x-2)(x+2)} \) and \( \frac{x}{x+2} \) must include all factors from both denominators. The LCD is \[ (x - 2)(x + 2). \]

Step 3: Rewrite Fractions with LCD

Both fractions need to be rewritten with the least common denominator. The first fraction \( \frac{1}{(x-2)(x+2)} \) is already in terms of the LCD. For the second fraction \( \frac{x}{x+2} \), multiply both the numerator and denominator by \( (x-2) \): \[ \frac{x(x-2)}{(x+2)(x-2)}. \]

Step 4: Combine the Fractions

Now that both fractions have the same denominator, they can be combined: \[ \frac{1}{(x-2)(x+2)} + \frac{x(x-2)}{(x+2)(x-2)} = \frac{1 + x(x-2)}{(x-2)(x+2)}. \]

Step 5: Simplify the Numerator

Expand and simplify the numerator: \[ 1 + x(x-2) = 1 + x^2 - 2x. \]

Step 6: Final Simplification

After expansion, the expression becomes: \[ \frac{x^2 - 2x + 1}{(x-2)(x+2)}. \] This is the simplified expression, as the numerator does not factor further to cancel with the denominator.

Key Concepts

Factoring

Factoring is a crucial step in dealing with rational expressions. It enables us to rewrite expressions in simpler, more manageable forms. In this problem, before any calculation, we notice a difference of squares in the first fraction's denominator: \( x^2 - 4 \). This pattern is pretty common and can be factored as \( (x - 2)(x + 2) \). By recognizing and factoring this expression, we can prepare the ground for further steps such as finding the least common denominator.

Factoring helps identify parts of expressions that can simplify or cancel out later. By factoring out terms, we can often see underlying structures and components that guide us to solution strategies. In this case, factoring reduces complexity and aligns the expressions for the next steps.

Least Common Denominator

Finding the least common denominator (LCD) is essential when dealing with fractions, especially when you need to combine them. The LCD is essentially the smallest expression that both denominators can fit into evenly.

In our problem, we have the denominators \( (x - 2)(x + 2) \) and \( x + 2 \). To find the LCD, we must take all unique factors from each denominator. Since \( (x - 2) \) appears only in the first expression, we must include it to cover both fractions. Thus, the LCD is:

• \( (x - 2)(x + 2) \).

Determining the LCD is crucial for rewriting each fraction so that they have the same denominator, which will allow us to add them together in the next steps.

Combining Fractions

Once you have fractions with the same denominator, combining them is straightforward. You simply merge the numerators over the common denominator.

In this exercise, both fractions are rewritten with the LCD \( (x - 2)(x + 2) \). We adjust the second fraction \( \frac{x}{x+2} \) to match this common denominator by multiplying both its numerator and denominator by \( (x-2) \). This step ensures alignment with the first fraction:

• First fraction: \( \frac{1}{(x-2)(x+2)} \)
• Adjusted second fraction: \( \frac{x(x-2)}{(x+2)(x-2)} \)

With matching denominators, we combine these into one fraction:

• \( \frac{1 + x(x-2)}{(x-2)(x+2)} \).

This setup allows for direct simplification in the next step.

Simplifying Expressions

In the final step, the goal is to simplify the expression as much as possible. This involves working out any operations in the numerator and checking if further simplification or factor cancellation is possible.

For this problem, we focus on expanding and simplifying the numerator of the combined fraction:

• Initially: \( 1 + x(x-2) \)
• After expansion: \( x^2 - 2x + 1 \)

This polynomial doesn't allow for further factoring that would cancel with the denominator, indicating that the expression is already as simple as it will get in this context.

Simplifying expressions might not always lead to a simpler form visually, but it's necessary to reach the most reduced version possible, revealing its true mathematical representation.