Problem 9 - College Algebra with Modeling and Visualization

Question

Simplify the expression. $$ -\frac{z+2}{4 z+8} $$

Step-by-Step Solution

Answer

The simplified expression is \(-\frac{1}{4}\).

Step 1: Identify the Common Factors

Look at the denominator \(4z + 8\) and determine if there is a common factor that can be factored out. Notice that both terms have a factor of 4. Thus, we can rewrite the denominator as \(4(z + 2)\).

Step 2: Simplify the Fraction

Substitute the simplified denominator into the expression: \(-\frac{z+2}{4(z+2)}\). In this form, it's visible that \(z+2\) appears in both the numerator and the denominator.

Step 3: Cancel the Common Factors

Since \((z+2)\) is a common factor in both the numerator and the denominator, we can cancel it out. This simplifies the expression to \(-\frac{1}{4}\), as any non-zero quantity divided by itself is 1.

Key Concepts

Common Factors

Finding common factors is an essential part of simplifying expressions. Think of common factors like the common ingredients two recipes might have. By "factoring out" these shared ingredients, we can simplify what's cooked up.

In the case of mathematical expressions, common factors in the numerator and the denominator allow us to simplify fractions more easily. In our problem, we noticed that both terms in the denominator share a number as a factor: 4. This is identified by looking for numbers or variables that can evenly divide into each term without leaving a remainder.

Recognizing these common factors helps us to break down complex expressions into simpler forms that are easier to work with.

Numerator and Denominator

The terms 'numerator' and 'denominator' are part of the language of fractions. The numerator is the top part of a fraction, and it answers 'how many?' or 'what is to be divided?'. The denominator, sitting below the "fraction bar," tells you 'into how many parts?' or 'what is divided?'

Generically, in a fraction \( \frac{a}{b} \), \( a \) is the numerator and \( b \) is the denominator. Understanding the roles of these parts helps us in simplification. It’s like understanding the parts of a recipe: one part is what you need, and the other is how it’s arranged or shared.

In our specific problem, \( z+2 \) is the numerator and \( 4z+8 \) is the denominator. Recognizing these allows us to focus on simplifying based on the structure and components of the fraction.

Fraction Simplification

Simplifying fractions is akin to tidying up a room; removing the clutter makes it easier to enjoy and useful. Simplification involves reducing a fraction to its simplest form, making it straightforward and efficient for further calculations or interpretations.

In our given expression, \( -\frac{z+2}{4z+8} \), we achieve simplification by canceling out common factors identified in both the numerator and the denominator.

• First, we rewrite the denominator with its factored common part: \( 4(z + 2) \).
• Then, recognizing that \( z + 2 \) is common in both parts, we can "cancel" it out.

This step leaves us with \( -\frac{1}{4} \) which is the simplest form of our original fraction. Thus, simplification is a process of identifying, factoring, and eliminating unnecessary complexities, arriving at an easy-to-understand result.