Problem 8 - Elementary Algebra

Question

For the following problems, reduce each rational expression to lowest terms. $$ \frac{6}{3 x-12} $$

Step-by-Step Solution

Answer

Question: Reduce the rational expression to lowest terms: \(\frac{6}{3x-12}\) Answer: \(\frac{2}{x-4}\)

Step 1: Identify the numerator and the denominator

The given rational expression is: $$ \frac{6}{3x-12} $$ Here, the numerator is 6 and the denominator is \(3x-12\).

Step 2: Factor the numerator and the denominator

We need to factor the numerator and the denominator to find the GCF. So we have: Numerator: \(6 = 2 \times 3\) Denominator: \(3x-12 = 3(x-4)\)

Step 3: Find the GCF and divide by it

The GCF of the numerator and the denominator is 3. Now, divide both the numerator and the denominator by the GCF: $$ \frac{6}{3x-12} = \frac{2 \times 3}{3(x-4)} = \frac{2}{x-4} $$

Step 4: Write the simplified expression

The simplified expression is: $$ \frac{2}{x-4} $$

Key Concepts

Factorization

Factorization is a process where you break down an expression into simpler parts, known as factors, that when multiplied together give the original expression. This is essential in simplifying rational expressions. Take the rational expression \( \frac{6}{3x-12} \). To simplify, we look at the denominator \(3x-12\). Factorizing this gives us \(3(x-4)\). By doing so, we can easily identify common factors in the numerator and denominator, which helps in reducing the expression to its simplest form.

Think of factorization as breaking down a complex puzzle into smaller, manageable pieces; it's a crucial step if we want to simplify the algebraic fraction effectively.

Greatest Common Factor (GCF)

The Greatest Common Factor (GCF), also known as the Greatest Common Divisor (GCD), is the largest number that divides two or more numbers. In algebra, this concept extends to terms in expressions, where the GCF is the highest expression that divides both the numerator and denominator without a remainder. In our exercise, the GCF of \(6\) and \(3x-12\) is \(3\), since both can be divided by \(3\) to leave simpler terms.

Finding the GCF is like finding the biggest piece of a pie that will fit evenly into several other pies - once you find it, you can serve up the whole set in equally sized pieces, or in mathematical terms, simplify the expression.

Algebraic Fractions

Algebraic fractions are simply fractions that contain algebraic expressions, both in the numerator and the denominator. Much like regular fractions, we can simplify them by cancelling out common factors. However, with algebraic expressions, factorization becomes an essential tool. Taking our main example, \( \frac{6}{3x-12} \), the objective is to reduce it to the simplest form by eliminating common factors. It's important to always ensure that the expressions you cancel are truly common factors to avoid altering the value of the fraction.

Reducing Fractions to Lowest Terms

Reducing fractions to their lowest terms means making the fraction as simple as possible. This is done by dividing both the numerator and the denominator by their GCF. For the fraction \( \frac{6}{3x-12} \), once we've factorized the denominator to \(3(x-4)\) and identified \(3\) as the GCF, we divide both terms by \(3\), which simplifies our fraction to \( \frac{2}{x-4} \).

This process is akin to minimizing baggage; you want to carry only what is essential. Similarly, when we reduce fractions to their lowest terms, we strip down to the most basic form without changing its value, allowing for easier manipulation and understanding of the expression.