How To Factor The Expression 4x - 12?

Factoring expressions is a fundamental skill in algebra, and it's crucial for simplifying equations, solving problems, and understanding more advanced mathematical concepts. In this article, we will walk through the process of factoring the expression 4x - 12 step by step. By the end of this guide, you will not only understand how to factor this particular expression but also grasp the general principles of factoring that can be applied to a variety of algebraic problems. We will delve into the concept of the greatest common factor (GCF), illustrate how to identify it, and demonstrate how to use it to rewrite the expression in a factored form. Additionally, we will compare the factored form with the provided options to determine the correct answer. Factoring is not just a mathematical exercise; it's a way of rewriting expressions to reveal their underlying structure, making them easier to work with and understand. Whether you're a student learning algebra for the first time or someone looking to refresh your skills, this guide will provide you with a clear and comprehensive approach to factoring. So, let's dive in and master the art of factoring!

Understanding the Basics of Factoring

Before we jump into factoring the expression 4x - 12, let's make sure we understand the basic principles of factoring. Factoring, in essence, is the reverse process of distribution. When we distribute, we multiply a term across an expression within parentheses. For example, if we have 4(x - 3), we distribute the 4 to both terms inside the parentheses, resulting in 4x - 12. Factoring, on the other hand, involves taking an expression like 4x - 12 and rewriting it as a product of its factors, such as 4(x - 3). The goal of factoring is to identify the common elements or factors that can be extracted from each term in the expression. These factors can be numbers, variables, or a combination of both. Identifying these common factors is a crucial step, and it's where the concept of the greatest common factor (GCF) comes into play. The GCF is the largest factor that divides evenly into all terms of the expression. Finding the GCF simplifies the factoring process and ensures that the expression is factored completely. In our example, the GCF of 4x and -12 is 4, which we will see in detail later. Understanding these basic principles is crucial for successfully factoring not only this expression but also more complex expressions you'll encounter in algebra. So, keep these concepts in mind as we move forward with the factoring process.

Identifying the Greatest Common Factor (GCF)

To factor the expression 4x - 12, the first and most crucial step is to identify the greatest common factor (GCF). The GCF is the largest number or expression that divides evenly into all terms of the given expression. In our case, the expression is 4x - 12, which consists of two terms: 4x and -12. To find the GCF, we need to consider the coefficients and the variables separately. First, let's look at the coefficients, which are the numerical parts of the terms. We have 4 and -12. The factors of 4 are 1, 2, and 4, while the factors of 12 are 1, 2, 3, 4, 6, and 12. The common factors of 4 and 12 are 1, 2, and 4. The greatest among these common factors is 4. Next, we consider the variables. The term 4x has a variable x, while the term -12 does not have any variable. Since there is no common variable in both terms, the variable part of the GCF is simply 1. Therefore, the GCF of 4x and -12 is the greatest common coefficient, which is 4. Once we've identified the GCF, we can proceed with the factoring process. This step is essential because factoring out the GCF simplifies the expression and makes it easier to work with. Without identifying the GCF correctly, the factoring might not be complete, and the expression might not be in its simplest form.

Factoring out the GCF from 4x - 12

Now that we have identified the greatest common factor (GCF) of 4x - 12 as 4, the next step is to factor it out. Factoring out the GCF means dividing each term in the expression by the GCF and rewriting the expression as a product of the GCF and the remaining terms inside parentheses. In our case, we will divide both terms, 4x and -12, by the GCF, which is 4. Let's start with the first term, 4x. When we divide 4x by 4, we get x. This is because 4x ÷ 4 = x. Next, we divide the second term, -12, by 4. This gives us -3, as -12 ÷ 4 = -3. Now that we have divided each term by the GCF, we can rewrite the expression in factored form. We write the GCF, which is 4, outside the parentheses, and the results of the divisions inside the parentheses. So, the factored form of 4x - 12 is 4(x - 3). This means that we have rewritten the original expression as a product of 4 and the expression (x - 3). Factoring out the GCF simplifies the expression and reveals its underlying structure. It is a crucial step in solving equations and simplifying algebraic expressions. By factoring out the GCF, we have successfully transformed 4x - 12 into its factored form, which is 4(x - 3). This form is much easier to work with in many algebraic contexts.

Verifying the Factored Form

After factoring an expression, it's always a good practice to verify the factored form to ensure that we have done it correctly. This step helps to catch any errors and provides confidence in our answer. To verify the factored form, we can use the distributive property, which is the reverse of factoring. In our case, we factored the expression 4x - 12 and obtained the factored form 4(x - 3). To verify this, we will distribute the 4 back into the parentheses. Distributing 4 into (x - 3) means multiplying 4 by both terms inside the parentheses. So, we have 4 * x and 4 * -3. Multiplying 4 by x gives us 4x, and multiplying 4 by -3 gives us -12. Combining these results, we get 4x - 12, which is the original expression we started with. Since distributing the factored form gives us back the original expression, we can be confident that our factoring is correct. This verification step is crucial, especially when dealing with more complex expressions. It ensures that we have accurately factored the expression and haven't made any mistakes along the way. By verifying the factored form, we gain assurance in our solution and can proceed with further steps in problem-solving or simplification with confidence.

Comparing with the Given Options

Now that we have factored the expression 4x - 12 and obtained the factored form 4(x - 3), the next step is to compare our result with the given options. This comparison will help us identify the correct answer among the choices provided. The given options are:

A. 4(x - 8) B. 4(x - 4) C. 4(x - 6) D. 4(x - 3)

By comparing our factored form, 4(x - 3), with these options, it is clear that option D, 4(x - 3), matches our result exactly. The other options, 4(x - 8), 4(x - 4), and 4(x - 6), have different constants inside the parentheses, which indicates that they are not the correct factored forms of the expression 4x - 12. This comparison step is crucial in multiple-choice questions, as it allows us to confirm our solution and choose the correct answer with confidence. By carefully comparing our result with the options, we can avoid selecting an incorrect answer due to a minor error in the factoring process. In this case, the comparison clearly shows that option D is the correct answer, as it is identical to the factored form we derived. This final step solidifies our understanding and ensures that we have correctly factored the given expression.

Conclusion: The Correct Answer

In conclusion, we have successfully factored the expression 4x - 12 by identifying the greatest common factor (GCF), factoring it out, and verifying the factored form. We found that the GCF of 4x and -12 is 4. By dividing each term by the GCF and rewriting the expression, we obtained the factored form 4(x - 3). To ensure the accuracy of our factoring, we distributed the 4 back into the parentheses, which resulted in the original expression, 4x - 12. This verification step confirmed that our factored form is correct. Finally, we compared our factored form with the given options and found that option D, 4(x - 3), matches our result perfectly. Therefore, the correct answer to the question of factoring the expression 4x - 12 is D. 4(x - 3). This process demonstrates the importance of understanding the basic principles of factoring, identifying the GCF, and verifying the solution to ensure accuracy. Factoring is a fundamental skill in algebra, and mastering it is crucial for solving more complex problems. By following these steps, you can confidently factor various algebraic expressions and enhance your mathematical abilities.

Therefore, the final answer is:

D. 4(x - 3)