Hannah's Factoring Errors Analysis Of 24xy + 15y

Factoring expressions is a fundamental skill in algebra, allowing us to simplify complex equations and solve for variables more efficiently. When factoring, it's crucial to identify the greatest common factor (GCF) and correctly apply the distributive property. In this article, we will dissect Hannah's attempt to factor the expression $24xy + 15y$, pinpointing the errors in her approach and providing a comprehensive explanation to ensure a clear understanding of the correct method. We'll break down each step, highlighting the misconceptions and offering a step-by-step guide to accurate factoring.

Understanding the Problem: Factoring 24xy + 15y

The initial problem presented is the expression $24xy + 15y$. The goal is to factor this expression, which means rewriting it as a product of its factors. This involves identifying the greatest common factor (GCF) of the terms and then expressing the original expression as the product of the GCF and the remaining factors. Factoring is a critical skill in algebra, enabling simplification of expressions, solving equations, and understanding underlying mathematical structures. When done correctly, factoring provides an equivalent expression that is often more manageable and insightful. A common mistake is overlooking the complete factorization or miscalculating the GCF, which leads to an incorrect result. To master factoring, it is important to have a solid understanding of the distributive property and the concept of common factors. Errors in factoring can propagate through subsequent steps in solving a problem, making it essential to develop a methodical and precise approach. Understanding the nuances of factoring helps in various mathematical contexts, including polynomial manipulation, equation solving, and calculus. The expression $24xy + 15y$ is a binomial, meaning it has two terms. Both terms share common factors, which makes it factorable. Accurately factoring this expression requires careful consideration of both numerical coefficients and variable components. In this case, the coefficients are 24 and 15, and the variable component y is common to both terms. The process of factoring involves dividing out the GCF from each term and expressing the original binomial as a product. This is the reverse operation of distribution, and understanding this inverse relationship is crucial for mastering factoring. Factoring not only simplifies expressions but also helps in identifying roots of equations and understanding the behavior of functions. It is a cornerstone of algebraic manipulation and is applied extensively in higher mathematics. To efficiently factor the given expression, we need to identify the largest common factor of both the numerical and variable parts, ensuring that the resulting factors are in their simplest form. This process is fundamental in simplifying complex algebraic expressions and solving various mathematical problems.

Step 1: Identifying the Greatest Common Factor (GCF)

Identifying the greatest common factor (GCF) is the first and most crucial step in factoring any expression. The GCF is the largest factor that divides evenly into all terms of the expression. In the case of $24xy + 15y$, we need to consider both the numerical coefficients (24 and 15) and the variable parts ($xy$ and $y$). The GCF will be the product of the largest number that divides both coefficients and the highest power of the variable(s) common to all terms. For the numerical coefficients, the factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24, while the factors of 15 are 1, 3, 5, and 15. The largest number that appears in both lists is 3, making it the numerical part of the GCF. When it comes to variables, we look for the variables that are present in all terms. Here, both terms have y, but only the first term has x. Thus, y is the variable part of the GCF. Combining the numerical and variable parts, we find that the GCF of $24xy$ and $15y$ is $3y$. This means that $3y$ is the largest expression that can divide both $24xy$ and $15y$ without leaving a remainder. Failing to identify the correct GCF will lead to an incomplete factorization, which is a common mistake in algebra. Students sometimes pick a smaller common factor, leaving further factorization steps necessary. Accurate identification of the GCF ensures that the factorization is complete and in its simplest form. Factoring out the GCF simplifies the original expression, making it easier to work with in subsequent algebraic manipulations. Mastering this step is vital for solving equations, simplifying expressions, and understanding the relationships between factors and multiples. The GCF not only simplifies the terms but also reveals the underlying structure of the expression, which is critical in advanced mathematical concepts. Therefore, careful consideration and accurate computation of the GCF are essential for successful factoring. Properly identifying the GCF is a foundational step that simplifies the entire factoring process and ensures the accuracy of the final result.

Step 2: Dividing by the GCF and Finding Remaining Factors

Once the greatest common factor (GCF) is correctly identified, the next step involves dividing each term in the original expression by the GCF. This process reveals the remaining factors that will form the other part of the factored expression. In our case, the GCF of $24xy + 15y$ is $3y$. We now need to divide each term, $24xy$ and $15y$, by $3y$. When we divide $24xy$ by $3y$, we perform the division on both the numerical coefficients and the variable parts separately. $24$ divided by $3$ is $8$, and $xy$ divided by $y$ leaves $x$ (since the $y$'s cancel out). Thus, $24xy / 3y = 8x$. Similarly, when we divide $15y$ by $3y$, $15$ divided by $3$ is $5$, and $y$ divided by $y$ is $1$. Therefore, $15y / 3y = 5$. These resulting quotients, $8x$ and $5$, are the terms that will be inside the parentheses in the factored expression. It's crucial to perform this division accurately, as any mistake here will lead to an incorrect factorization. This step directly applies the distributive property in reverse, which is the foundation of factoring. The quotients obtained represent how the original expression can be reconstructed by distributing the GCF back into the parentheses. Careful attention to detail and a solid understanding of division are necessary to avoid errors in this step. The process of dividing by the GCF not only simplifies the expression but also reveals the relationship between the terms and their common factor. This understanding is vital for more complex factoring problems and algebraic manipulations. Proper division ensures that the remaining terms are in their simplest form and that the factored expression is equivalent to the original one. Therefore, this step is a critical bridge between identifying the GCF and correctly writing the factored expression. Dividing each term by the GCF correctly is a cornerstone of accurate factoring, setting the stage for a precise and simplified representation of the original expression.

Step 3: Writing the Factored Expression

After dividing each term by the greatest common factor (GCF), the final step is to write the expression in its factored form. This involves placing the GCF outside the parentheses and the quotients obtained from the division inside the parentheses. In our example, the GCF of $24xy + 15y$ is $3y$. We divided $24xy$ by $3y$ and got $8x$, and we divided $15y$ by $3y$ and got $5$. These are the terms that will be inside the parentheses. The factored expression is thus written as $3y(8x + 5)$. This means that the original expression $24xy + 15y$ is equivalent to $3y$ multiplied by the binomial $(8x + 5)$. The distributive property can be used to verify the correctness of the factored expression. If we distribute $3y$ back into the parentheses, we should obtain the original expression. Specifically, $3y * 8x = 24xy$ and $3y * 5 = 15y$, so $3y(8x + 5) = 24xy + 15y$, confirming that our factored expression is correct. This step consolidates the previous steps and represents the original expression in a simplified, factored form. Factoring is essentially the reverse of distribution, and understanding this relationship is crucial for verifying the accuracy of the factored result. The factored form often provides deeper insights into the structure of the expression and is useful in solving equations and simplifying algebraic manipulations. Properly writing the factored expression ensures that all factors are accounted for and that the expression is represented in its most simplified form. This step is the culmination of the factoring process, transforming the original expression into a product of its factors. Therefore, careful attention to detail and a solid understanding of the distributive property are essential for accurately completing this step and achieving a correct factored expression.

Analyzing Hannah's Errors

Reviewing Hannah's work, we can pinpoint the errors in her approach to factoring the expression $24xy + 15y$. Hannah correctly identified the greatest common factor (GCF) as $3y$, which is a good start. However, the subsequent steps reveal a critical misunderstanding of how to write the factored expression. Hannah's first step, correctly stating that the GCF $(24xy, 15y) = 3y$, demonstrates an understanding of how to find the common factors of the terms. The second step, where she divides each term by the GCF ($24xy / 3y = 8x$ and $15y / 3y = 5$), is also performed correctly. This indicates that Hannah understands the process of dividing each term by the GCF to find the remaining factors. The error lies in the third step: $3(24xy + 15y)$. This step incorrectly multiplies the GCF by the original expression instead of writing the GCF multiplied by the result of the division. Hannah seems to have missed the crucial step of placing the quotients inside the parentheses. Instead, she simply wrote the GCF multiplied by the original expression, which is not a valid factoring step. The correct factored form should be $3y(8x + 5)$, where $8x$ and $5$ are the quotients obtained in the second step. This error indicates a misunderstanding of the distributive property in reverse, which is the basis of factoring. Hannah's mistake highlights the importance of not only identifying the GCF but also correctly expressing the original expression as the product of the GCF and the remaining factors. The final expression Hannah wrote is not equivalent to the original expression, which means the factoring is incorrect. Analyzing this error is crucial for understanding the common pitfalls in factoring and developing a systematic approach to avoid them. Hannah's mistake underscores the need for a clear understanding of the factoring process as a whole, rather than just individual steps. This analysis emphasizes the importance of careful attention to detail and a thorough understanding of the underlying principles of factoring to prevent similar errors.

Correcting Hannah's Work: The Right Approach

To correct Hannah's work and factor the expression $24xy + 15y$ accurately, we need to follow a systematic approach. The first step, as Hannah correctly identified, is to find the greatest common factor (GCF) of the terms $24xy$ and $15y$. The GCF is the largest factor that divides evenly into both terms. The factors of $24$ are $1, 2, 3, 4, 6, 8, 12,$ and $24$, while the factors of $15$ are $1, 3, 5,$ and $15$. The largest number common to both lists is $3$. Both terms also share a common variable, $y$. Therefore, the GCF is $3y$. Next, we divide each term in the original expression by the GCF. Dividing $24xy$ by $3y$ gives us $8x$, since $24 / 3 = 8$ and $xy / y = x$. Dividing $15y$ by $3y$ gives us $5$, since $15 / 3 = 5$ and $y / y = 1$. Now, we write the factored expression by placing the GCF outside the parentheses and the quotients inside the parentheses. This gives us $3y(8x + 5)$. This expression means that $24xy + 15y$ can be written as the product of $3y$ and $(8x + 5)$. To verify the correctness of our factored expression, we can distribute $3y$ back into the parentheses. $3y * 8x = 24xy$ and $3y * 5 = 15y$, so $3y(8x + 5) = 24xy + 15y$. This confirms that our factored expression is equivalent to the original expression. The correct factored form of $24xy + 15y$ is $3y(8x + 5)$. This step-by-step approach ensures that we accurately factor the expression and avoid the errors Hannah made. Emphasizing the distributive property in reverse is key to understanding and performing factoring correctly. This correction not only provides the right answer but also reinforces the correct method, highlighting the importance of following a logical and systematic process in algebraic manipulations. Thus, by accurately identifying the GCF, dividing each term by the GCF, and correctly writing the factored expression, we can ensure the accurate factorization of $24xy + 15y$.

Key Takeaways for Accurate Factoring

To ensure accurate factoring, it is essential to grasp several key concepts and apply a systematic approach. First and foremost, understanding the concept of the greatest common factor (GCF) is paramount. The GCF is the largest factor that divides evenly into all terms of the expression, and accurately identifying it is the foundation of correct factoring. This involves considering both the numerical coefficients and the variable parts of the terms. A common mistake is overlooking a common factor, resulting in incomplete factorization. Secondly, the process of dividing each term by the GCF must be performed meticulously. This step reveals the remaining factors that will form the other part of the factored expression. Accurate division is critical, as errors here will lead to an incorrect factored form. The distributive property in reverse is the guiding principle behind this step, and a strong understanding of division is necessary to avoid mistakes. Next, writing the factored expression correctly involves placing the GCF outside the parentheses and the quotients obtained from the division inside the parentheses. This step consolidates the previous steps and expresses the original expression as a product of its factors. A common error is misplacing or omitting terms, as seen in Hannah's work. Finally, verifying the correctness of the factored expression by distributing the GCF back into the parentheses is a crucial step. If the result matches the original expression, the factoring is correct. This step acts as a check and balance, ensuring accuracy and preventing errors from propagating through subsequent steps. Consistent practice and attention to detail are essential for mastering factoring. Common mistakes can be avoided by following a systematic approach and carefully reviewing each step. Factoring is not just a mechanical process; it requires a deep understanding of the underlying principles and relationships between factors and multiples. By internalizing these key takeaways and adopting a methodical approach, one can significantly improve their factoring skills and achieve accurate results consistently. Accurate factoring is a cornerstone of algebraic proficiency, crucial for simplifying expressions, solving equations, and understanding advanced mathematical concepts.

By analyzing Hannah's errors and outlining the correct approach, this article provides a comprehensive understanding of factoring the expression $24xy + 15y$. The key lies in correctly identifying the GCF, dividing each term by the GCF, and writing the factored expression accurately. Consistent practice and attention to detail will help in mastering this fundamental algebraic skill.