Question

Perform the indicated operation and simplify the result. Leave your answer in factored form. $$ \frac{x}{x^{2}-4}+\frac{1}{x} $$

Short answer

\( \frac{2(x^2 - 2)}{x(x + 2)(x - 2)} \)

Step-by-step solution

Identify the problem

The problem is to perform the arithmetic operation and simplify the expression: \( \frac{x}{x^2 - 4} + \frac{1}{x} \). The final answer should be left in factored form.

Factor the denominator

First, notice that \( x^2 - 4 \) is a difference of squares, which can be factored as \( (x + 2)(x - 2) \).

Rewrite the expression with the factored denominator

Rewrite the original problem using the factored form of the denominator: \( \frac{x}{(x + 2)(x - 2)} + \frac{1}{x} \).

Find a common denominator

The common denominator for the fractions \( \frac{x}{(x + 2)(x - 2)} \) and \( \frac{1}{x} \) is \( x(x + 2)(x - 2) \).

Rewrite each fraction with the common denominator

Rewrite each fraction with the common denominator: \( \frac{x \times x}{x(x + 2)(x - 2)} + \frac{1 \times (x + 2)(x - 2)}{x(x + 2)(x - 2)} \). This simplifies to \( \frac{x^2}{x(x + 2)(x - 2)} + \frac{(x + 2)(x - 2)}{x(x + 2)(x - 2)} \).

Combine the fractions

Now combine the fractions: \( \frac{x^2 + (x + 2)(x - 2)}{x(x + 2)(x - 2)} \).

Simplify the numerator

Simplify the numerator: \( (x + 2)(x - 2) = x^2 - 4 \). Thus the expression becomes \( \frac{x^2 + x^2 - 4}{x(x + 2)(x - 2)} \).

Combine like terms

Combine like terms in the numerator: \( x^2 + x^2 - 4 = 2x^2 - 4 \). The expression now is \( \frac{2x^2 - 4}{x(x + 2)(x - 2)} \).

Factor the numerator

Factor out the greatest common factor of the numerator, which is 2: \( \frac{2(x^2 - 2)}{x(x + 2)(x - 2)} \).

Simplify and leave in factored form

Leave the final answer in factored form: \( \frac{2(x^2 - 2)}{x(x + 2)(x - 2)} \).

Key concepts

Factoring

Factoring is a method used to simplify algebraic expressions. It involves breaking down a complex expression into simpler factors that, when multiplied together, give the original expression. In our exercise, we factored the denominator \(x^2 - 4\) into \((x + 2)(x - 2)\). This step is crucial because it allows us to identify a common denominator, making it easier to combine the fractions. Factoring helps in simplifying the expressions, which is essential before performing any operations on them.

Common Denominator

Finding a common denominator is a key step when adding or subtracting fractions. The common denominator is a shared multiple of the denominators of the fractions involved. In our problem, the fractions \( \frac{x}{(x + 2)(x - 2)} \) and \( \frac{1}{x} \) have different denominators. By finding a common denominator, we make it possible to combine the two fractions into a single fraction. For our exercise, the common denominator is \( x(x + 2)(x - 2)\). This common denominator allows us to rewrite each fraction so that they can be combined easily.

Algebraic Fractions

Algebraic fractions are fractions where the numerators and/or denominators contain algebraic expressions. Working with algebraic fractions requires careful handling, especially when it comes to combining and simplifying them. In algebraic fractions, we apply the same basic principles of fractions but with algebraic terms. For instance, in this exercise, we handled fractions such as \( \frac{x}{x^2 - 4} \) and \( \frac{1}{x} \) by factoring, finding common denominators, and simplifying them just like we do with numerical fractions.

Simplifying Expressions

Simplifying expressions involves reducing an expression to its simplest form. This can include factoring, combining like terms, and canceling out common factors in the numerator and denominator. In our problem, we simplified the combined fraction by factoring out the greatest common factor from the numerator, which is 2. We started with the expression \( \frac{2x^2 - 4}{x(x + 2)(x - 2)} \) and factored out a 2 from the numerator to get \( \frac{2(x^2 - 2)}{x(x + 2)(x - 2)} \). Simplifying expressions is crucial for making complex math problems more manageable and readable.