Polynomial Addition Error Analysis Lucias Mistake Explained

Lucia's attempt to simplify the sum of two polynomials, specifically $(3x^2 + 3x + 5) + (7x^2 - 9x + 8)$, resulted in an incorrect answer of $10x^2 - 12x + 13$. To unravel the mystery behind this algebraic misstep, we must meticulously examine each step of the polynomial addition process. Polynomial addition, at its core, involves combining like terms – terms that share the same variable and exponent. In this case, we need to group the $x^2$ terms, the $x$ terms, and the constant terms separately.

Identifying the Culprit: A Step-by-Step Analysis

The first step in solving this problem is to correctly identify and combine like terms. Like terms are those that have the same variable raised to the same power. In the given problem, we have quadratic terms ($x^2$), linear terms ($x$), and constant terms. To correctly add the polynomials, we must add the coefficients of the like terms.

Let's break down the addition:

• Quadratic Terms: The quadratic terms are $3x^2$ and $7x^2$. Adding their coefficients, we get $3 + 7 = 10$. Thus, the combined quadratic term should be $10x^2$.
• Linear Terms: The linear terms are $3x$ and $-9x$. Adding their coefficients, we get $3 + (-9) = -6$. Therefore, the combined linear term should be $-6x$.
• Constant Terms: The constant terms are $5$ and $8$. Adding them, we get $5 + 8 = 13$.

Therefore, the correct sum of the polynomials should be $10x^2 - 6x + 13$. Comparing this to Lucia's answer of $10x^2 - 12x + 13$, we can pinpoint the error: Lucia incorrectly combined the linear terms. She arrived at $-12x$ instead of the correct $-6x$.

The Heart of the Matter: Miscombining Linear Terms

Lucia's mistake lies in the combination of the linear terms. The problem presents the linear terms as $3x$ and $-9x$. The correct arithmetic operation involves adding these terms, which translates to adding their coefficients: $3 + (-9)$. This sum is $-6$, not $-12$. Lucia's answer suggests she might have incorrectly performed the operation, perhaps by subtracting the coefficients instead of adding them, or by making a simple arithmetic error. This underscores the importance of careful attention to signs when performing algebraic operations.

It's possible that Lucia might have confused the rules of addition with those of multiplication or subtraction, particularly when dealing with negative coefficients. For instance, a common mistake is to treat the addition of a negative number as a separate subtraction operation, leading to errors in the final result. Understanding the fundamental rules of arithmetic and their application in algebraic expressions is crucial for avoiding such pitfalls.

Distinguishing Sum from Difference: A Crucial Concept

While Lucia's primary error was in combining the linear terms, it's important to address the other options presented. Option A suggests she found the difference instead of the sum. If Lucia had calculated the difference, she would have subtracted the second polynomial from the first: $(3x^2 + 3x + 5) - (7x^2 - 9x + 8)$. This operation would yield a different result altogether, specifically: $-4x^2 + 12x - 3$. This result is significantly different from Lucia's answer, indicating that she did attempt to add the polynomials, but made an error within the addition process itself. Therefore, option A is not the primary error Lucia committed, although understanding the difference between sum and difference is a fundamental concept in algebra.

Unpacking Polynomial Addition: The Significance of Like Terms

Option B suggests that Lucia combined the quadratic terms ($3x^2$ and $7x^2$) incorrectly. However, her answer shows the correct combination of these terms: $10x^2$. This indicates that Lucia understood the concept of combining like terms and correctly applied it to the quadratic terms. Therefore, the error does not stem from miscombining the $x^2$ terms. Understanding how to combine like terms is a cornerstone of polynomial arithmetic. It's essential to recognize that only terms with the same variable and exponent can be combined. For example, $3x^2$ and $7x^2$ are like terms because they both have the variable $x$ raised to the power of 2. However, $3x^2$ and $3x$ are not like terms because they have different exponents.

Deciphering the Mistake: A Focus on Linear Terms

Option C directly points to the source of the error: Lucia combined the terms $3x$ and $-9x$ incorrectly. As we've established, the correct sum of these terms is $-6x$, while Lucia obtained $-12x$. This confirms that the error lies within the manipulation of the linear terms. This reinforces the idea that attention to detail, particularly with signs and coefficients, is paramount in algebraic manipulations. Even a small arithmetic error can lead to a significantly different final result.

Solidifying Understanding: Avoiding Future Errors

To avoid similar errors in the future, Lucia (and anyone learning algebra) should focus on the following:

• Careful Attention to Signs: Pay close attention to the signs (positive or negative) of the coefficients when adding or subtracting terms. A misplaced sign can drastically alter the result.
• Systematic Organization: Organize the terms by their degree (exponent) to ensure that like terms are easily identified and combined. This can be particularly helpful when dealing with more complex polynomials.
• Double-Checking Work: Always double-check your work, especially the arithmetic involved in adding or subtracting coefficients. Simple errors can be easily overlooked if the work is not reviewed carefully.
• Practice, Practice, Practice: The more you practice polynomial addition and subtraction, the more comfortable and confident you will become with the process. This will also help you identify and avoid common errors.

The Final Verdict: Option C is the Key

In conclusion, Lucia's error was in combining the linear terms $3x$ and $-9x$ incorrectly. She obtained $-12x$ instead of the correct $-6x$. This highlights the importance of paying close attention to signs and arithmetic when performing algebraic operations. While understanding the difference between sum and difference and correctly combining like terms are crucial concepts, the specific error in this case lies in the miscalculation of the linear term coefficients. Therefore, the correct answer is C. She combined the terms $3x$ and $-9x$ incorrectly.

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