Problem 40 - Intermediate Algebra

Question

Simplify each rational expression. $$ \frac{6 x-12}{x^{2}-4} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{6}{x + 2}\), assuming \(x \neq 2\).

Step 1: Factor the Numerator

The numerator of the expression \(6x - 12\) can be factored by taking out the greatest common factor (GCF), which is 6: \(6x - 12 = 6(x - 2)\).

Step 2: Factor the Denominator

The denominator \(x^2 - 4\) is a difference of squares and can be factored as \((x - 2)(x + 2)\).

Step 3: Write the Factored Expression

Now, substitute the factored forms into the expression. We have: \(\frac{6(x - 2)}{(x - 2)(x + 2)}\).

Step 4: Cancel Common Factors

Identify and cancel the common factor \((x - 2)\) in the numerator and the denominator: \(\frac{6(x - 2)}{(x - 2)(x + 2)} = \frac{6}{x + 2}\), assuming \(x eq 2\).

Key Concepts

Factoring Polynomials

Factoring polynomials is a fundamental skill in algebra that involves breaking down a polynomial into a product of simpler polynomials. In the given exercise, both the numerator and the denominator of the rational expression needed to be factored. This step helps in simplifying the expression by revealing common factors that can be canceled.

When factoring polynomials, always look for a greatest common factor (GCF) first. This is the largest expression that is a factor of all terms in the polynomial. Identifying and factoring out the GCF is the first step in simplifying complex expressions. In the numerator \(6x - 12\), the GCF is 6, so it can be factored as \(6(x-2)\).

By continuously practicing factoring, you can become familiar with patterns like the difference of squares, perfect square trinomials, and other special factoring techniques. It also develops skills in recognizing how to express polynomials in their simplest forms.

Greatest Common Factor

The greatest common factor (GCF) is the largest factor that divides each term in a polynomial expression. Finding the GCF is a crucial first step in the process of factoring a polynomial.

To determine the GCF, identify the smallest power of all common variables involved and the greatest whole number that divides the coefficients. In our example, the polynomial in the numerator is \(6x - 12\). The GCF of 6 and 12 is 6, as it is the largest number that can divide both without leaving a remainder. We can then factor out the GCF from the entire expression, which gives us \(6(x-2)\).

Factoring out the GCF simplifies expressions and makes them easier to work with, especially when simplifying rational expressions. Always begin by scanning your polynomial for the GCF to streamline the simplification process.

Difference of Squares

The difference of squares is a special factorization method applicable when you have an expression of the format \(a^2 - b^2\). This can be rewritten as \((a-b)(a+b)\). This method is useful in simplifying expressions that might otherwise seem complex.

In the denominator of the original exercise, \(x^2 - 4\) is an example of a difference of squares, as it can be broken down into \((x - 2)(x + 2)\) because \(4\) can be represented as \(2^2\).

This factoring technique is particularly beneficial in rational expressions, like in the step-by-step solution, as it allows for canceling out terms. It further simplifies the rational expression by identifying common factors with those in the numerator. Hence, it's one of the first patterns to check for when tackling polynomial expressions.