Problem 67 - Elementary Algebra

Question

For exercises \(35-86\), simplify. $$ \frac{x}{x-2}-\frac{4 x}{x^{2}-4} $$

Step-by-Step Solution

Answer

The simplified expression is \[ \frac{x}{x+2} \] for \[ x eq 2 \].

Step 1 - Factor the Denominator

First, notice that the denominator in the second term can be factored. The expression \[ x^2 - 4 \] is a difference of squares and can be written as \[ (x-2)(x+2).\]

Step 2 - Rewrite the Expression

Rewrite the given expression by replacing \[ x^2 - 4 \] with \[ (x-2)(x+2). \] The expression now looks like this: \[ \frac{x}{x-2} - \frac{4x}{(x-2)(x+2)}. \]

Step 3 - Find a Common Denominator

To combine the fractions, we need a common denominator. Here, the common denominator is \[ (x-2)(x+2). \]

Step 4 - Rewrite the First Fraction

Rewrite the first fraction to have the common denominator by multiplying the numerator and denominator by \[ (x+2). \] The fraction then becomes: \[ \frac{x(x+2)}{(x-2)(x+2)} = \frac{x^2 + 2x}{(x-2)(x+2)}. \]

Step 5 - Combine the Fractions

Now combine the fractions over the common denominator: \[ \frac{x^2 + 2x}{(x-2)(x+2)} - \frac{4x}{(x-2)(x+2)} = \frac{x^2 + 2x - 4x}{(x-2)(x+2)}. \]

Step 6 - Simplify the Numerator

Combine like terms in the numerator: \[ \frac{x^2 + 2x - 4x}{(x-2)(x+2)} = \frac{x^2 - 2x}{(x-2)(x+2)}. \]

Step 7 - Factor the Numerator

Factor out an \[ x \] from the numerator: \[ \frac{x(x - 2)}{(x-2)(x+2)}. \]

Step 8 - Simplify the Fraction

Cancel the common factor of \[ (x-2) \] in the numerator and denominator, resulting in: \[ \frac{x}{x+2}. \] Make sure \[ x eq 2 \] to avoid division by zero.

Key Concepts

Factoring Polynomials

To simplify algebraic expressions, factoring polynomials is fundamental. Factoring involves breaking down a complicated polynomial into the product of simpler polynomials. One common method is recognizing patterns like the difference of squares. For example, \text{\[ x^2 - 4 \]} neatens into \text{\[ (x-2)(x+2) \]}, making it easier to work with the expression. Factoring also aids in identifying and canceling out common terms when simplifying fractions. Recognizing such patterns and practicing their use is key to mastering polynomial operations.

Common Denominators

When dealing with fractions, a common denominator is essential for combining them. Essentially, a common denominator is a shared multiple of the denominators of the fractions involved. In our exercise, we saw the need to find a common denominator for \text{\[ \frac{x}{x-2} - \frac{4x}{(x-2)(x+2)} \]}, which was identified as \text{\[ (x-2)(x+2) \]}. By rewriting fractions with this common denominator, you can combine them as you would with whole numbers, making it simpler to perform subsequent arithmetic operations on the numerators.

Combining Fractions

Combining fractions involves unifying multiple fractional expressions into one. This requires a consistent denominator across all fractions involved. Once a common denominator is found, rewrite each fraction so they share this new denominator. Next, perform the necessary operations on the numerators while keeping the common denominator intact. In our example, rewriting \text{\[ \frac{x}{x-2} \]} as \text{\[ \frac{x(x+2)}{(x-2)(x+2)} \]} allowed the expression to merge with \text{\[ \frac{4x}{(x-2)(x+2)} \]}. Finally, subtract the numerators for the simplified result.

Difference of Squares

Recognizing the difference of squares allows for quick simplification in polynomial expressions. A difference of squares takes the form \text{\[ a^2 - b^2 = (a-b)(a+b) \]}. In our case, \text{\[ x^2 - 4 \]} becomes \text{\[ (x-2)(x+2) \]}. Identifying and using the difference of squares makes it easier to factor and reduce fractions. It's useful in many algebraic problems, including simplifying complex equations and solving polynomial equations.