Problem 14 - A Graphical Approach to College Algebra

Question

Write each rational expression in lowest terms. $$\frac{-8(y-4)}{(y+2)(y-4)}$$

Step-by-Step Solution

Answer

The expression simplifies to \(-\frac{8}{y+2}\).

Step 1: Identify Common Factors

Look for common factors both in the numerator and the denominator. The given expression is \(\frac{-8(y-4)}{(y+2)(y-4)}\). Here, \((y-4)\) is a common factor in both the numerator and the denominator.

Step 2: Cancel the Common Factor

Since \((y-4)\) is found in both the numerator and the denominator, we can cancel it out. The expression simplifies to \(-8/(y+2)\) as the \((y-4)\) factors cancel each other.

Step 3: Express in Lowest Terms

After canceling out the common factor, the expression \(-8/(y+2)\) is already in its simplest form. There are no further common factors between \(-8\) and \(y+2\).

Key Concepts

Lowest Terms

When working with rational expressions, expressing them in "lowest terms" means simplifying the expression so that the numerator and denominator have no common factors other than 1. This is very similar to simplifying fractions in basic arithmetic, where the goal is to break down the expression to its most basic form. A rational expression is in its lowest terms when it cannot be reduced further. In the exercise given, we simplify the expression to \(-\frac{8}{y+2}\) which indicates that there are no remaining factors common to both the top and bottom. Thus, the expression is in its lowest terms.

Common Factors

"Common factors" play a crucial role in simplifying rational expressions. These are elements that are shared between the numerator and the denominator. By identifying and canceling these common factors, we reduce the expression. In the original exercise, the term \((y-4)\) was present in both the numerator and the denominator.

The process involves:

• Identifying the factors in the numerator.
• Identifying the factors in the denominator.
• Spotting any common factors.
• Canceling the common factors to simplify the expression.

Recognizing common factors simplifies expressions, making calculations much easier and less error-prone.

Simplification

"Simplification" is the process where we reduce expressions by eliminating factors that appear both in the numerator and denominator. Simplifying a rational expression makes it easier to work with by displaying the expression in an easier form without changing its value. For example, reducing \(\frac{-8(y-4)}{(y+2)(y-4)}\) to \(-\frac{8}{y+2}\) involves removing the common factor \((y-4)\).

Steps in simplification include:

• Identifying factors.
• Canceling common factors.
• Writing the simplified expression.

A simplified expression is more manageable and usually more aesthetically pleasing.

Numerator and Denominator

The "numerator and denominator" are two essential parts of any fraction or rational expression. The numerator is the top part, while the denominator is the bottom part. When working with rational expressions like \(\frac{-8(y-4)}{(y+2)(y-4)}\), it's crucial to understand what each part represents, as this understanding aids in proper simplification.

In our rational expression:

• The numerator is \(-8(y-4)\), representing the part being divided.
• The denominator is \((y+2)(y-4)\), representing the divisor.

Understanding these components and their relationships facilitates the simplification process, leading to accurately reduced expressions.