Factoring 3x + 24y A Step-by-Step Explanation

Factoring is a fundamental concept in algebra, allowing us to simplify expressions and solve equations more easily. When we factor an expression, we are essentially rewriting it as a product of its factors. This process involves identifying common factors within the terms of the expression and extracting them. In this article, we will delve into the process of factoring the algebraic expression 3x + 24y, breaking down each step to ensure a clear understanding. We will explore the concepts of the greatest common factor (GCF), the distributive property, and how they play a crucial role in factoring. By the end of this discussion, you will not only be able to factor this specific expression but also gain the skills to tackle a wide range of factoring problems. Mastering these techniques is essential for success in algebra and beyond, as factoring is a cornerstone of many advanced mathematical concepts. The ability to factor expressions accurately and efficiently will significantly enhance your problem-solving abilities in various mathematical contexts. So, let's begin our exploration of factoring and unlock the secrets behind simplifying algebraic expressions. This journey will equip you with the knowledge and skills to confidently approach factoring challenges, making your mathematical endeavors more manageable and rewarding. Understanding the underlying principles of factoring is key to building a strong foundation in algebra, paving the way for more advanced topics such as solving equations, simplifying rational expressions, and working with polynomials. By focusing on clarity and step-by-step explanations, we aim to make the process of factoring accessible and enjoyable for learners of all levels. So, get ready to embark on this mathematical adventure and discover the power of factoring! We'll start by identifying the greatest common factor, a crucial first step in simplifying the given expression.

Identifying the Greatest Common Factor (GCF)

To factor the expression 3x + 24y, the first step is to identify the greatest common factor (GCF) of the coefficients and variables. The greatest common factor is the largest number that divides evenly into both terms. In this case, the terms are 3x and 24y. We need to find the GCF of the coefficients 3 and 24. The factors of 3 are 1 and 3. The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24. Comparing the factors, we see that the greatest common factor of 3 and 24 is 3. Now, let's consider the variables. The first term has the variable 'x', and the second term has the variable 'y'. Since they are different variables and do not share any common variable factors, the GCF for the variables is 1. Therefore, the greatest common factor for the entire expression 3x + 24y is 3. Identifying the GCF is a crucial step because it allows us to simplify the expression effectively. Once we've determined the GCF, we can use it to factor out the common term, making the expression easier to work with. This process is not just about finding a number; it's about understanding the structure of the expression and how we can rewrite it in a more simplified form. By carefully examining the coefficients and variables, we can pinpoint the GCF and pave the way for the next step in factoring. Remember, the GCF is the largest factor that divides both terms, so taking the time to find it accurately is essential for successful factoring. This foundation will enable us to proceed with confidence and complete the factoring process with ease. The concept of GCF extends beyond simple expressions; it's a fundamental tool in algebra and number theory, applicable in various contexts such as simplifying fractions and solving equations. So, mastering the art of identifying the GCF is a valuable skill that will serve you well in your mathematical journey. With the GCF identified, we are now ready to move on to the next step: factoring out the GCF from the expression.

Factoring Out the GCF

After identifying the GCF as 3, we can now factor it out from the expression 3x + 24y. Factoring out the GCF involves dividing each term in the expression by the GCF and writing the expression as a product of the GCF and the resulting quotient. We begin by dividing the first term, 3x, by the GCF, 3. This gives us x, since 3x / 3 = x. Next, we divide the second term, 24y, by the GCF, 3. This gives us 8y, since 24y / 3 = 8y. Now, we can rewrite the original expression 3x + 24y as the product of the GCF, 3, and the sum of the quotients we just calculated, which are x and 8y. This gives us the factored form: 3(x + 8y). The process of factoring out the GCF is based on the distributive property in reverse. The distributive property states that a(b + c) = ab + ac. In our case, we are going from the form ab + ac to the form a(b + c). By factoring out the GCF, we are essentially undoing the distribution. This is a powerful technique for simplifying expressions and solving equations. The factored form, 3(x + 8y), is equivalent to the original expression, 3x + 24y, but it is written in a more compact and useful form. This form makes it easier to analyze the expression, solve equations involving it, and perform other algebraic manipulations. Factoring out the GCF is a critical skill in algebra, and mastering it will significantly improve your ability to work with algebraic expressions. It's a foundational technique that you'll use repeatedly in more advanced topics. The ability to recognize and factor out the GCF is essential for simplifying expressions, solving equations, and understanding the structure of algebraic relationships. This step-by-step process of dividing each term by the GCF and then rewriting the expression as a product is a fundamental concept in algebra. By practicing this technique, you'll develop a strong intuition for factoring and be able to apply it confidently in various mathematical contexts. With the expression now factored, we have successfully transformed it into a product of its factors, which often makes it easier to work with in further calculations or problem-solving scenarios.

Verifying the Factored Form

To ensure our factored form is correct, we can use the distributive property to expand it and see if we arrive back at the original expression. The factored form we obtained is 3(x + 8y). To expand this, we multiply the 3 by each term inside the parentheses. Multiplying 3 by x gives us 3x. Multiplying 3 by 8y gives us 24y. So, expanding the factored form 3(x + 8y) results in 3x + 24y, which is indeed the original expression. This verification step is crucial because it confirms that our factoring process was accurate and that the factored form is equivalent to the original expression. It's a way of checking our work and ensuring that we haven't made any mistakes along the way. The distributive property is a fundamental concept in algebra, and it plays a key role in both factoring and expanding expressions. Understanding how to use the distributive property effectively is essential for success in algebra. By expanding the factored form, we are essentially reversing the factoring process, and if we arrive back at the original expression, we can be confident that our factoring is correct. This verification step not only confirms our answer but also reinforces our understanding of the relationship between factoring and expanding expressions. It's a valuable habit to develop, as it helps to minimize errors and build confidence in our mathematical abilities. Verifying the factored form is a simple yet powerful technique that can save us from making mistakes and ensure that our solutions are accurate. This step is particularly important when dealing with more complex expressions, where the chances of making an error are higher. By taking the time to verify our work, we can ensure that our solutions are reliable and that we have a solid understanding of the underlying concepts. Remember, accuracy is paramount in mathematics, and verifying our results is a key step in achieving accuracy. With the verification complete, we have confirmed that our factored form is correct, and we can confidently move on to using it in further calculations or problem-solving scenarios. This process of factoring and verifying not only helps us simplify expressions but also deepens our understanding of algebraic manipulations and their underlying principles. The ability to both factor and expand expressions is a valuable skill that will serve you well in your mathematical journey, enabling you to tackle a wide range of problems with confidence and precision.

Conclusion: The Factored Form of 3x + 24y

In conclusion, the factored form of the expression 3x + 24y is 3(x + 8y). We arrived at this answer by identifying the greatest common factor (GCF) of the terms 3x and 24y, which is 3. We then factored out this GCF by dividing each term by 3 and writing the expression as a product of the GCF and the resulting quotient. This process demonstrates the power of factoring in simplifying algebraic expressions. Factoring allows us to rewrite expressions in a more compact and manageable form, making them easier to work with in various mathematical contexts. The ability to factor expressions is a fundamental skill in algebra, and it's essential for solving equations, simplifying fractions, and performing other algebraic manipulations. Throughout this discussion, we have emphasized the importance of understanding the underlying concepts of factoring, such as the GCF and the distributive property. These concepts are not only crucial for factoring simple expressions like 3x + 24y but also for tackling more complex factoring problems. By mastering the techniques of factoring, you will gain a significant advantage in your mathematical studies. Factoring is a versatile tool that can be applied in a wide range of situations, from solving basic equations to simplifying complex algebraic expressions. The process of factoring involves breaking down an expression into its constituent factors, which can provide valuable insights into the structure and properties of the expression. This understanding is essential for developing a deeper appreciation of algebra and its applications. The factored form 3(x + 8y) provides a concise representation of the original expression, highlighting the common factor of 3 and the relationship between the terms x and 8y. This factored form can be used to simplify calculations, solve equations, and gain a better understanding of the expression's behavior. By consistently practicing factoring techniques and applying them in various contexts, you will develop a strong intuition for factoring and be able to approach factoring problems with confidence and efficiency. Factoring is not just a mathematical skill; it's a way of thinking that can enhance your problem-solving abilities in various domains. The ability to identify patterns, break down complex problems into simpler components, and express them in a more manageable form is a valuable asset in any field. So, embrace the challenge of factoring, and you will unlock a powerful tool for mathematical exploration and problem-solving. This journey into factoring has equipped you with the knowledge and skills to confidently tackle similar problems and delve deeper into the world of algebra.

Therefore, the correct answer is A. 3(x+8y)