Problem 36 - Algebra for College Students

Question

Reduce each rational expression to its lowest terms. $$\frac{7 x-14}{7 x}$$

Step-by-Step Solution

Answer

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

Step 1: Identify Common Factors

First, recognize the common factors in both the numerator and the denominator of the rational expression. The numerator is \(7x - 14\) and the denominator is \(7x\).

Step 2: Factor the Numerator

Factor out the greatest common factor (GCF) in the numerator. The GCF of \(7x - 14\) is 7. So, \(7x - 14\) can be factored as \(7(x - 2)\).

Step 3: Rewrite the Expression

Rewrite the original expression using the factored form of the numerator: $$\frac{7(x - 2)}{7x}$$

Step 4: Cancel Common Factors

Cancel the common factor of 7 from both the numerator and the denominator: $$\frac{7(x - 2)}{7x} = \frac{x - 2}{x}$$

Step 5: Simplify

The simplified form of the rational expression after canceling the common factors is: \(\frac{x-2}{x}\)

Key Concepts

Common Factors

To simplify rational expressions, the first step is identifying common factors in the numerator and the denominator. A common factor is a term that can be evenly divided into each term of a polynomial. For instance, in the expression \(\frac{7x - 14}{7x}\), we have a common factor of 7 between the numerator and the denominator. Recognizing these common factors helps to simplify the expressions significantly. This process involves looking for terms that can be factored out from both parts of the fraction. By splitting and simplifying these terms, we make the expression easier to manage.

Factoring

Factoring is an essential skill in reducing rational expressions. It involves breaking down a polynomial into the product of its factors. In our example, let's look at the numerator \((7x - 14)\). To factor this, we find the greatest common factor (GCF). In this case, it's 7. Thus, we can rewrite \(7x - 14\) as \(7(x - 2)\). By doing this, we simplify the expression and lay the groundwork for canceling out common terms. Factoring is not limited to extracting numerical coefficients; it can also involve variables within the polynomial.

Rational Expressions

A rational expression is a fraction where both the numerator and the denominator are polynomials. Simplifying these expressions follows similar rules to simplifying numerical fractions. Our goal is to rewrite them in the simplest form by canceling out common factors. In the expression \(\frac{7x - 14}{7x}\), after factoring the numerator, we can cancel the common factor of 7. This leaves us with \(\frac{x - 2}{x}\). Simplifying rational expressions helps to better understand their behavior and makes them more manageable when solving equations.

Numerator and Denominator

Understanding the roles of the numerator and the denominator is crucial in working with rational expressions. The numerator is the top part of the fraction and the denominator is the bottom part. In our example \(\frac{7x - 14}{7x}\), \ (7x - 14) \ is the numerator and \ (7x) \ is the denominator. Simplification involves manipulating these two components to their lowest terms by factoring out common elements. Learning to effectively identify and factor both the numerator and denominator helps in achieving the final simplified form, \(\frac{x - 2}{x}\), which is much simpler to work with in higher-level math problems.