Problem 35 - Intermediate Algebra

Question

Simplify each rational expression. $$ \frac{4 x^{2}}{2 x^{3}-12 x^{2}} $$

Step-by-Step Solution

Answer

\(\frac{2}{x - 6}\) is the simplified expression.

Step 1: Identify the Greatest Common Factor (GCF)

Identify the greatest common factor of the terms in the numerator and the denominator. In this case, for the numerator \(4x^2\), there are no other terms. However, in the denominator \(2x^3 - 12x^2\), the GCF is \(2x^2\).

Step 2: Factor Out GCF in Denominator

Factor the GCF out of the denominator. The expression \(2x^3 - 12x^2\) can be rewritten as:\[2x^2(x - 6)\]

Step 3: Rewrite the Rational Expression

Replace the original denominator with the factored form:\[\frac{4x^2}{2x^2(x - 6)}\]

Step 4: Simplify by Canceling Common Factors

Cancel the common factor \(2x^2\) present in both the numerator and the denominator. Thus, the expression simplifies:\[\frac{4x^2}{2x^2(x - 6)} = \frac{2}{x - 6}\]

Step 5: Final Simplified Expression

The expression after simplification is:\[\frac{2}{x - 6}\]

Key Concepts

Greatest Common Factor

Finding the greatest common factor (GCF) is the first step when simplifying rational expressions. The GCF is the largest factor shared by all the terms in an expression. Locating it can make simplification much easier. Let's break it down in terms of an example:The expression provided is \(\frac{4x^2}{2x^3 - 12x^2}\). Here, the numerator is a simple term, \(4x^2\), which doesn't require factoring. However, the denominator is a binomial, \(2x^3 - 12x^2\), which can be more complex.

• First, focus on the coefficients: 2 and 12. Their GCF is 2 since it's the largest number that evenly divides both.
• Next, look at the variables: the lowest power of \(x\) in each term is \(x^2\). Therefore, \(x^2\) is the GCF for the variable part.

Now, combining the results from these steps, the GCF of the full expression \(2x^3 - 12x^2\) is \(2x^2\). Understanding the GCF helps in reducing terms later when simplifying the expression.

Factoring

Factoring is the next crucial process. Once you've determined the greatest common factor, your goal is to take it out of each term in the expression so that what remains can be easily simplified or canceled. With our denominator \(2x^3 - 12x^2\), let's factor it using the GCF we computed, which is \(2x^2\). By dividing each term inside the polynomial by \(2x^2\), we successfully factor it:

• \(2x^3\) divided by \(2x^2\) gives \(x\).
• \(-12x^2\) divided by \(2x^2\) gives \(-6\).

Thus, we rewrite the denominator as \(2x^2(x - 6)\). This means we've expressed the original problem in a form where part of the expression can be canceled, simplifying the whole rational expression. Factoring is a pivotal practice that helps reveal common factors and allows us to create simpler expressions.

Simplifying Rational Expressions

Simplifying rational expressions means reducing them to their simplest form. This involves canceling out common factors found in both the numerator and denominator once the expression has been factored. In our worked example, after factoring the denominator into \(2x^2(x - 6)\), the rational expression becomes \(\frac{4x^2}{2x^2(x - 6)}\). Now, both the numerator and the denominator contain a common factor of \(2x^2\).

• Cancelling \(2x^2\) from both the numerator and the denominator yields a much simpler expression.
• This results in \(\frac{2}{x - 6}\), significantly simplifying our original expression.

Simplifying rational expressions is all about clarity. It helps in better understanding and working with expressions involving variables and provides a cleaner form which is often easier to use in further mathematical operations.