Problem 58 - Algebra and Trigonometry

Question

Perform the addition or subtraction and simplify. $$ \frac{x}{x^{2}-4}+\frac{1}{x-2} $$

Step-by-Step Solution

Answer

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

Step 1: Identify the Denominators

The given expression is \( \frac{x}{x^2 - 4} + \frac{1}{x-2} \). The first fraction has a denominator \( x^2 - 4 \), which can be factored as \((x-2)(x+2)\). The second fraction has the denominator \( x-2 \). We need a common denominator to perform the addition.

Step 2: Find the Common Denominator

To find a common denominator, we need to include all unique factors from each denominator. The common denominator is \((x-2)(x+2)\).

Step 3: Rewrite Each Fraction with the Common Denominator

Rewrite the first fraction (\( \frac{x}{(x-2)(x+2)} \)) with the common denominator, which it already has. Rewrite the second fraction by multiplying both the numerator and the denominator by \((x+2)\): \( \frac{1(x+2)}{(x-2)(x+2)} = \frac{x+2}{(x-2)(x+2)} \).

Step 4: Add the Fractions

Combine the two fractions: \( \frac{x}{(x-2)(x+2)} + \frac{x+2}{(x-2)(x+2)} \). Add the numerators since the denominators are the same: \( \frac{x + (x + 2)}{(x-2)(x+2)} \).

Step 5: Simplify the Numerator

Simplify the numerator: \( x + x + 2 = 2x + 2 \).

Step 6: Factor and Simplify the Expression

Factor the numerator: \( 2x + 2 = 2(x + 1) \). The expression becomes \( \frac{2(x + 1)}{(x-2)(x+2)} \). This is the simplest form of the expression as there are no common factors between the numerator and denominator.

Key Concepts

Addition of Fractions

When adding fractions, the first step is to ensure that both fractions have the same denominator. This process is essential because you can only add the numerators directly when the denominators are the same. In our given problem, the fractions are:

• \( \frac{x}{x^2 - 4} \)
• \( \frac{1}{x-2} \)

The denominator of the first fraction is a polynomial \( x^2 - 4 \), which can be factored further. The second fraction has a simple denominator \( x-2 \). Our task is to combine these by finding a common denominator, which will allow us to seamlessly add the fractions together.

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of simpler polynomials. In this exercise, the polynomial \( x^2 - 4 \) can be factored using the difference of squares method, a common technique in algebra. Applying this method to \( x^2 - 4 \), we realize:

• \( x^2 - 4 = (x-2)(x+2) \)

By breaking the polynomial into these factors, we make it easier to identify a common denominator for the fractions. Factoring is a fundamental skill for working with rational expressions, as it simplifies the expressions and reveals hidden relationships between terms.

Common Denominator

Finding a common denominator is a crucial step when adding fractions with different denominators. In our expression, after factoring \( x^2 - 4 \), we recognize the potential common denominator as \((x-2)(x+2)\). The second fraction's denominator \(x-2\) needs to be adjusted by including the factor \(x+2\) to match the first fraction. To do this, we multiply both the numerator and the denominator of the second fraction by \(x+2\), obtaining:

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

Now, both fractions share the common denominator \((x-2)(x+2)\), allowing us to easily add their numerators together.

Simplification of Expressions

Simplification is the process of reducing an expression to its simplest form. After adding fractions, we reach \( \frac{2x + 2}{(x-2)(x+2)} \). We can simplify the numerator \(2x + 2\) by factoring out common terms:

• \(2x + 2 = 2(x + 1)\)

Thus, our expression becomes \( \frac{2(x + 1)}{(x-2)(x+2)} \). We check for any common factors between the numerator and the denominator that we might further cancel, ensuring the expression is at its simplest form. Simplification enhances clarity and makes further operations, if necessary, easier to manage.