Problem 37 - Elementary and Intermediate Algebra

Question

Find the GCF of each list of terms. $$ 4(x+7), 9(x+7) $$

Step-by-Step Solution

Answer

The GCF is \((x+7)\).

Step 1: Identify the common factor

Examine both expressions \(4(x+7)\) and \(9(x+7)\) to identify if there is a common factor. Notice that both expressions have the factor \((x+7)\) in common.

Step 2: Determine the GCF

Since both terms share the same factor \((x+7)\), the greatest common factor (GCF) is \((x+7)\).

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division. They can be as simple as a single number or variable, or more complex with multiple terms and operations. In this context, an algebraic expression might look like

• Single term: \(3a\)
• Multiple terms: \(4x + 5\)

Understanding algebraic expressions is crucial, as it lays the groundwork for more advanced topics in algebra. For the given problem, the expressions were \(4(x+7)\) and \(9(x+7)\). Each is an example of an algebraic expression where grouping is used to represent repeated terms, making it simpler to manage.

Factoring

Factoring involves breaking down an expression into simpler parts that can be multiplied together to get the original expression. It's a powerful tool in algebra because it can simplify expressions and help solve equations. A common factoring technique is to look for the greatest common factor (GCF) of the terms, which is

• the largest factor that divides each term in the expression without leaving a remainder
• in the given problem, the expressions \(4(x+7)\) and \(9(x+7)\) both have the phrase \((x+7)\) as a common factor, making it the GCF.

By factoring out the common term, calculations can be simplified, making them easier to work with.

Mathematics Education

The goal of mathematics education is to equip students with tools and skills to solve real-world problems, develop logical reasoning, and cultivate a strong foundation in mathematical concepts. Understanding concepts such as algebraic expressions and factoring is essential because they appear frequently in more advanced math topics. Effective math education should focus on:

• Conceptual understanding, so students grasp why techniques work.
• Practice, which helps reinforce these concepts.
• Applications of math in everyday life, which shows students the relevance of what they're learning.

By emphasizing these areas, educators aim to inspire confidence and interest in mathematics, paving the way for lifelong learning and problem-solving skills.