Problem 29 - Elementary Algebra

Question

Factor out the GCF. $$ 4 x-8 $$

Step-by-Step Solution

Answer

The expression factors to \(4(x - 2)\).

Step 1: Identify the Common Factor

The expression given is \(4x - 8\). We need to determine the greatest common factor (GCF) of the terms. Both terms \(4x\) and \(-8\) share the factor \(4\), as it is the largest number that divides both evenly.

Step 2: Rewrite Using the GCF

Factor the GCF out of each term in the expression. Write the expression as: \(4(x) - 4(2)\).

Step 3: Factor Out the GCF

Factor the GCF \(4\) out of the entire expression: \(4(x - 2)\). This shows that both original terms are divisible by \(4\), giving us the factored form.

Key Concepts

Factoring Expressions

Factoring expressions is a crucial skill in algebra. It's the process of breaking down an expression into simpler terms, or factors, that multiply together to give the original expression. Think of it like taking apart a puzzle and then being able to see each piece clearly. This is particularly useful because it allows us to simplify expressions and solve equations more easily. Here, in the given exercise, you were asked to factor the expression \(4x - 8\).

The method begins with identifying elements that can be extracted or factored out to simplify the expression, making it more manageable to work with in algebraic operations. Factoring expressions also plays a pivotal role when working with quadratic equations or simplifying rational expressions. So, grasping factoring is essential for progressing in algebra. In practical terms, learning to factor helps in recognizing patterns and relationships between numbers and variables, enhancing comprehension in mathematical problem-solving.

Common Factors

The greatest common factor (GCF) is the biggest factor shared by two or more numbers or terms. It's like finding the secret ingredient that can be taken out of all the numbers involved. In the expression \(4x - 8\), the terms are \(4x\) and \(-8\). We look for common factors in these terms to simplify the expression.

Let's break it down:

• Identify the common factors among the coefficients (numbers in front of variables) of the terms. Here, both coefficients \(4\) and \(-8\) can be divided evenly by \(4\).
• Find if there's a common variable factor. For instance, both these terms have an \(x\), it would be considered a common factor if it were present in both.

By identifying \(4\) as the Greatest Common Factor for this specific problem, you're able to factor \(4\) out of each term to simplify to \(4(x - 2)\).

Remember, finding the common factor is as simple as determining the largest number that can neatly divide each term in the expression. It's about finding that 'biggest puzzle piece' shared among numbers or terms.

Algebra Basics

At its core, algebra is all about using symbols, usually letters, to represent numbers in equations and expressions. It's a language with its own rules and logic, designed to help us understand mathematical relationships.

One of the foundational aspects of learning algebra is understanding how to manipulate expressions and equations to simplify or solve them. This involves skills like factoring expressions and finding common factors, both of which were applicable in the example \(4x - 8\).

Some fundamental principles include:

• Using variables like \(x\) to represent unknown or generalized numbers.
• Understanding operations like addition, subtraction, multiplication, and division in the context of these variables.
• Recognizing patterns and relationships between numbers within an expression.

By equipping yourself with these basics, tackling more complex mathematical problems becomes far less daunting. The beauty of algebra is its ability to simplify real-world problems, making it a vital tool across various fields like engineering, economics, science, and more. Mastering these basics paves the way to success in more advanced mathematical studies.