Lucia's Error In Polynomial Addition Explained

In the realm of algebra, polynomial addition is a fundamental operation. It involves combining like terms to simplify expressions. A common mistake students make is incorrectly combining these terms, leading to an incorrect result. This article delves into a specific error encountered by Lucia while adding two polynomials. We'll analyze her steps, pinpoint the error, and reinforce the correct procedure for polynomial addition. This exploration will not only illuminate Lucia's mistake but also serve as a comprehensive guide for anyone seeking to master this essential algebraic skill. By understanding the nuances of combining like terms, readers can avoid similar pitfalls and confidently tackle polynomial addition problems. So, let's embark on this mathematical journey to unravel the mystery behind Lucia's error and solidify our understanding of polynomial addition.

Lucia was tasked with adding the polynomials ${(3x^2 + 3x + 5)}$ and ${(7x^2 - 9x + 8)}$. Her solution was ${10x^2 - 12x + 13}$. However, this answer is incorrect. The goal is to identify the specific error Lucia made during the addition process. We will break down the steps involved in polynomial addition and compare them to Lucia's work to pinpoint exactly where the mistake occurred. Understanding the nature of the error is crucial for learning and preventing similar mistakes in the future. This detailed analysis will help us not only correct Lucia's work but also gain a deeper understanding of the underlying principles of polynomial arithmetic. By carefully examining each step, we can uncover the error and reinforce the correct methods for adding polynomials.

Lucia's answer is ${10x^2 - 12x + 13}$, but the correct sum should be derived by carefully adding the coefficients of like terms. Let's dissect how Lucia might have arrived at this incorrect answer. First, let's consider the ${x^2}$ terms. Lucia likely correctly added ${3x^2}$ and ${7x^2}$ to get ${10x^2}$. This part of her solution is accurate. However, the error likely lies in how she handled the ${x}$ terms. To get ${-12x}$, she might have incorrectly subtracted the coefficients instead of adding them, or made a sign error. It's essential to remember that we are adding polynomials, so we must add the coefficients of the ${x}$ terms, considering their signs. We will analyze this further in the subsequent sections. Finally, the constant terms, 5 and 8, correctly add up to 13, suggesting she handled the constant terms appropriately. Therefore, the error most likely stems from the incorrect handling of the ${x}$ terms. By focusing on this specific part of the problem, we can pinpoint the exact mistake and learn how to avoid it in the future.

To correctly add the polynomials ${(3x^2 + 3x + 5)}$ and ${(7x^2 - 9x + 8)}$, we need to combine like terms. Like terms are those that have the same variable raised to the same power. In this case, we have ${x^2}$ terms, ${x}$ terms, and constant terms. Let's start with the ${x^2}$ terms: ${3x^2 + 7x^2 = 10x^2}$. Next, we combine the ${x}$ terms: ${3x + (-9x) = -6x}$. It's crucial to pay attention to the signs of the coefficients. Finally, we add the constant terms: ${5 + 8 = 13}$. Now, we combine these results to form the correct sum: ${10x^2 - 6x + 13}$. This systematic approach ensures that we accurately combine all like terms, avoiding the common errors associated with polynomial addition. By breaking down the problem into smaller steps, we minimize the chances of making mistakes and arrive at the correct solution. Understanding this process is vital for mastering polynomial arithmetic.

Comparing Lucia's answer, ${10x^2 - 12x + 13}$, with the correct sum, ${10x^2 - 6x + 13}$, reveals that the error lies in the ${x}$ term. Lucia obtained ${-12x}$, while the correct term is ${-6x}$. This indicates that she made a mistake when combining the coefficients of the ${x}$ terms, which are 3 and -9. To arrive at -12, she might have mistakenly subtracted 9 from -3 or perhaps added 3 and -9 but made a sign error in the calculation. The correct operation should be ${3 + (-9) = -6}$. This highlights the importance of carefully considering the signs when adding or subtracting coefficients. A simple sign error can lead to an incorrect result. By recognizing this specific mistake, we can emphasize the need for meticulous attention to detail when performing algebraic operations. Understanding where the error occurred allows us to learn from it and prevent similar errors in the future.

Lucia's error likely stems from a misunderstanding or misapplication of the rules for adding signed numbers. When adding the coefficients of the ${x}$ terms, 3 and -9, it's crucial to remember that adding a negative number is equivalent to subtraction. The correct operation is ${3 + (-9)}$, which is the same as ${3 - 9}$. A common mistake is to treat this as ${-9 - 3}$, which would result in -12. Another possibility is that Lucia correctly performed the subtraction but overlooked the sign of the larger number. The difference between 9 and 3 is 6, but since 9 is negative and has a larger absolute value, the result should be -6. This highlights the importance of understanding the number line and the rules for adding and subtracting integers. A strong foundation in these basic arithmetic principles is essential for success in algebra. By reviewing these concepts, students can avoid making similar mistakes in polynomial addition and other algebraic operations.

Therefore, the correct sum of the polynomials ${(3x^2 + 3x + 5)}$ and ${(7x^2 - 9x + 8)}$ is found by combining like terms: ${3x^2 + 7x^2 = 10x^2}$, ${3x + (-9x) = -6x}$, and ${5 + 8 = 13}$. Combining these terms gives us the final answer: ${10x^2 - 6x + 13}$. This result differs from Lucia's answer of ${10x^2 - 12x + 13}$ only in the coefficient of the ${x}$ term. This reinforces our conclusion that Lucia's error was specifically in the addition of the ${x}$ terms. The correct answer demonstrates the importance of careful attention to signs and the rules of arithmetic when performing algebraic operations. By following a systematic approach and double-checking each step, we can ensure accuracy and avoid common errors. This correct solution serves as a benchmark for understanding polynomial addition and identifying potential mistakes.

In conclusion, Lucia's error in adding the polynomials ${(3x^2 + 3x + 5)}$ and ${(7x^2 - 9x + 8)}$ was in correctly combining the ${x}$ terms. She arrived at ${-12x}$ instead of the correct ${-6x}$. This mistake likely stemmed from a misapplication of the rules for adding signed numbers, specifically overlooking the correct way to add 3 and -9. The correct sum of the polynomials is ${10x^2 - 6x + 13}$. This analysis highlights the crucial importance of paying close attention to the signs of coefficients and understanding the fundamental principles of arithmetic when performing algebraic operations. Polynomial addition, while seemingly straightforward, requires a meticulous approach to avoid common errors. By identifying and understanding these errors, students can strengthen their algebraic skills and confidently tackle more complex problems in the future. This example serves as a valuable lesson in the importance of precision and attention to detail in mathematics.