Factored Form Of 3x + 24y Step-by-Step Solution

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The question at hand involves finding the factored form of the algebraic expression 3x + 24y. Factoring is a fundamental concept in algebra, which essentially means breaking down an expression into its constituent parts, typically by identifying common factors. This process is the reverse of expansion, where we multiply terms together. In this article, we will delve deep into understanding the process of factorization, specifically focusing on the expression 3x + 24y, and provide a step-by-step guide to arrive at the correct factored form. We will not only solve the problem but also explore the underlying principles and techniques of factoring, ensuring a comprehensive understanding for readers of all levels.

To begin, let's understand why factoring is essential in algebra. Factoring simplifies complex expressions, making them easier to manipulate and solve. It's a crucial skill in solving equations, simplifying fractions, and various other algebraic operations. In the context of 3x + 24y, factoring will help us identify the common element that can be extracted from both terms, thereby simplifying the expression. This simplification can be particularly useful when dealing with more complex problems where this expression might be just one part of a larger equation or system.

The process of factoring involves identifying the greatest common factor (GCF) of the terms in the expression. The GCF is the largest number or expression that divides evenly into all terms. In our case, we have two terms: 3x and 24y. We need to find the largest factor that is common to both. To do this, we look at the coefficients (the numbers in front of the variables) and the variables themselves. The coefficient of the first term is 3, and the coefficient of the second term is 24. The factors of 3 are 1 and 3, while the factors of 24 include 1, 2, 3, 4, 6, 8, 12, and 24. Clearly, the greatest common factor of 3 and 24 is 3. Now, let’s examine the variables. The first term has 'x' and the second term has 'y'. Since they are different variables, there is no common variable factor in this case. Therefore, the GCF of the entire expression 3x + 24y is simply 3.

Once we have identified the GCF, the next step is to factor it out of the expression. This means dividing each term by the GCF and writing the expression as a product of the GCF and the resulting terms. In our example, we divide both 3x and 24y by 3. Dividing 3x by 3 gives us 'x', and dividing 24y by 3 gives us '8y'. Now, we write the factored form by placing the GCF outside the parentheses and the resulting terms inside the parentheses. This gives us 3(x + 8y). This factored form is equivalent to the original expression 3x + 24y, but it is written in a more simplified and useful manner.

To verify that our factoring is correct, we can use the distributive property to expand the factored form and see if it matches the original expression. The distributive property states that a(b + c) = ab + ac. Applying this to our factored form, 3(x + 8y), we multiply 3 by both 'x' and '8y'. This gives us 3 * x + 3 * 8y, which simplifies to 3x + 24y, the original expression. This confirms that our factored form, 3(x + 8y), is indeed correct. Understanding this verification process is crucial as it ensures accuracy and reinforces the relationship between factoring and expansion.

Detailed Solution and Explanation

To solve the problem of finding the factored form of 3x + 24y, we follow these steps:

  1. Identify the Greatest Common Factor (GCF): Look for the largest number and/or variable that divides evenly into all terms of the expression. In this case, the terms are 3x and 24y. The coefficients are 3 and 24. The GCF of 3 and 24 is 3. The variables are 'x' and 'y', which are different, so there is no common variable factor.
  2. Factor out the GCF: Divide each term in the expression by the GCF and write the expression as a product. Divide 3x by 3 to get 'x'. Divide 24y by 3 to get '8y'.
  3. Write the Factored Form: Place the GCF outside the parentheses and the resulting terms inside the parentheses. This gives us 3(x + 8y).
  4. Verification (Optional but Recommended): To ensure accuracy, expand the factored form using the distributive property. Multiply 3 by both 'x' and '8y': 3 * x + 3 * 8y = 3x + 24y, which matches the original expression.

Therefore, the factored form of 3x + 24y is 3(x + 8y). This process highlights the importance of identifying the GCF and correctly applying the distributive property in reverse to achieve factorization. Understanding these steps not only helps in solving this particular problem but also equips you with the skills to tackle more complex factoring problems in the future.

Analyzing the Options

Now, let's analyze the given options to determine the correct factored form of 3x + 24y.

  • A. 3(x + 8y): This is the correct factored form. As we demonstrated in the detailed solution, factoring out the GCF of 3 from 3x + 24y results in 3(x + 8y). This option accurately represents the factorization process and is the correct answer.
  • B. 3xy(x + 8y): This option is incorrect. It includes 'xy' as part of the factored form, which is not a common factor in the original expression. While 3 is a common factor, 'x' is only present in the first term and 'y' is only present in the second term. Therefore, 'xy' cannot be part of the GCF. Expanding this option would give us 3x²y + 24xy², which is not equivalent to the original expression 3x + 24y.
  • C. 3(3x + 24y): This option is also incorrect. Factoring out 3 again from the original expression would not simplify it; it would only multiply the expression by 3. This is not the correct application of factoring. Expanding this option would result in 9x + 72y, which is not the same as 3x + 24y.
  • D. 3xy(3x + 24y): This option is incorrect for similar reasons as option B. It incorrectly includes 'xy' as a common factor and also fails to correctly factor the original expression. Expanding this option would give us 9x²y + 72xy², which is significantly different from the original expression 3x + 24y.

Therefore, by carefully analyzing each option and comparing it to the correct factorization process, we can confidently conclude that option A, 3(x + 8y), is the only correct factored form of the given expression. This analysis reinforces the importance of understanding the fundamental principles of factoring and how to apply them accurately.

Common Mistakes to Avoid When Factoring

Factoring algebraic expressions is a crucial skill in mathematics, but it's also an area where students often make mistakes. To ensure accuracy and mastery, it's essential to be aware of these common pitfalls. When factoring, especially with expressions like 3x + 24y, there are several mistakes to watch out for:

  1. Not Identifying the Greatest Common Factor (GCF): One of the most frequent errors is failing to identify the correct GCF. As we discussed earlier, the GCF is the largest factor that divides evenly into all terms of the expression. In the case of 3x + 24y, the GCF is 3. Some students might overlook this and not factor at all, or they might identify a smaller common factor, leading to an incomplete factorization. Always ensure you've found the greatest common factor, not just a common factor.
  2. Incorrectly Factoring Variables: Another common mistake is misapplying variable factoring. For example, in the expression 3x + 24y, 'x' and 'y' are different variables, and there is no common variable factor. A mistake would be to include 'x' or 'y' or 'xy' in the factored form, as seen in options B and D. Remember, a variable can only be factored out if it is present in all terms of the expression.
  3. Distributing Instead of Factoring: Factoring is the reverse of distribution. A common error is to confuse the two processes. Factoring involves identifying common elements and extracting them, while distribution involves multiplying a term across an expression within parentheses. For instance, option C, 3(3x + 24y), demonstrates a misunderstanding of this concept. This option essentially multiplies the original expression by 3 rather than factoring it.
  4. Incomplete Factoring: Sometimes, students may identify a common factor but not factor completely. For example, if an expression can be factored further after an initial factorization, failing to do so would be an error. In the case of 3x + 24y, factoring out 3 is the complete factorization, but in more complex expressions, there might be multiple steps to achieve full factorization.
  5. Not Checking by Expanding: A simple yet effective way to avoid mistakes is to check your answer by expanding the factored form. Use the distributive property to multiply the factored terms and see if you arrive back at the original expression. If the expanded form doesn't match the original expression, there's an error in the factoring process. This step is crucial for verification and ensuring accuracy.

By being mindful of these common mistakes and practicing consistently, you can improve your factoring skills and avoid errors. Understanding the underlying principles and techniques of factoring is key to mastering this essential algebraic skill.

Conclusion

In conclusion, the factored form of the expression 3x + 24y is 3(x + 8y). This result is achieved by identifying the greatest common factor (GCF) of the terms, which is 3, and then factoring it out from the expression. This process involves dividing each term by the GCF and writing the expression as a product of the GCF and the resulting terms. The steps include identifying the GCF, factoring it out, writing the factored form, and verifying the result by expanding. Understanding the concepts of factoring, the distributive property, and common mistakes to avoid are crucial for mastering this fundamental algebraic skill.

Throughout this comprehensive guide, we have not only solved the problem but also explored the underlying principles and techniques of factoring. We have emphasized the importance of identifying the GCF, correctly applying the distributive property in reverse, and verifying the result to ensure accuracy. By avoiding common mistakes and practicing consistently, you can enhance your factoring skills and confidently tackle a wide range of algebraic problems. Factoring is a foundational concept in algebra, and a solid understanding of it will pave the way for success in more advanced mathematical topics. We hope this article has provided you with a clear and thorough understanding of factoring, empowering you to approach similar problems with confidence and precision.