Problem 77 - Algebra A Combined Function

Question

Factor each numerator and denominator. Then simplify if possible. $$ \frac{2 x-14}{2} $$

Step-by-Step Solution

Answer

The simplified expression is \(x - 7\).

Step 1: Factoring the Numerator

The numerator is \(2x - 14\). Notice that both terms have a common factor of 2. We can factor out the 2 to get: \(2(x - 7)\).

Step 2: Express the Fraction with Factored Numerator

After factoring the numerator, the expression becomes: \(\frac{2(x - 7)}{2}\).

Step 3: Cancel Common Factors

Since both the numerator and the denominator have a common factor of 2, we can cancel it out. This simplifies the fraction to \(x - 7\).

Key Concepts

Factoring

Factoring is a crucial skill in algebra, especially when dealing with rational expressions. It involves breaking down an expression into "factors" or components that multiply together to give the original expression. When we look at a polynomial, like the numerator in our exercise, we search for numbers or expressions that are common to all the terms.
We begin by identifying any common factors in our expression. In this exercise, the expression is \( 2x - 14 \). We can see that both terms, \( 2x \) and \( -14 \), share a factor of 2. Thus, we factor out 2, rewriting the expression as \( 2(x - 7) \).

• Identify common numerical factors and variables.
• Factor out completely to simplify further.

Factoring simplifies complex problems and is the first step towards making rational expressions easier to handle. Understanding how to find and factor out common elements is essential for simplifying and solving algebraic equations.

Simplifying Expressions

Simplifying expressions means making them as straightforward as possible. For rational expressions, this involves reducing the expression to its simplest form by factoring and canceling common terms in the numerator and the denominator.
When we simplify, we transform a complex expression into a simpler one without changing its value. In our case, we start with the fraction \( \frac{2x - 14}{2} \). After factoring the numerator to \( 2(x - 7) \), the expression becomes \( \frac{2(x - 7)}{2} \).
Now, both the numerator and the denominator have a factor of 2. By cancelling these factors, we obtain the simplified expression \( x - 7 \).

• Identify and factor common terms first.
• Cancel common factors to simplify.
• Always check that the expression cannot be simplified further.

Simplifying is key to solving algebraic problems efficiently and accurately, so mastering this skill can save a lot of time and effort in solving more complex equations.

Common Factors

Common factors are numbers or terms that appear in different parts of an algebraic expression. Identifying and using common factors is a primary step in the process of factoring and simplifying rational expressions.
In our exercise, the common factor is 2, which appears both in the numerator \(2(x - 7)\) and in the denominator 2. Spotting this allows us to simplify the fraction effectively.

• Search for the greatest common factor (GCF) among terms.
• Factor out the GCF to make simplification possible.

When working with rational expressions, recognizing common factors helps simplify expressions faster and makes solving equations easier. This concept forms a foundation for further algebraic manipulation, aiding in everything from solving equations to simplifying complex expressions.