What Mistake Did Lucia Make When She Calculated $\left(3x^2 + 3x + 5\right) + \left(7x^2 - 9x + 8\right) = 10x^2 - 12x + 13$?

Lucia encountered an error while performing polynomial addition, specifically when simplifying the expression $\left(3x^2 + 3x + 5\right) + \left(7x^2 - 9x + 8\right)$. Her result, $10x^2 - 12x + 13$, deviates from the correct simplification. This article delves into the step-by-step solution to pinpoint Lucia's mistake, offering a comprehensive guide to polynomial addition and error identification.

Understanding Polynomial Addition

Key Concepts

Polynomial addition involves combining like terms. Like terms are those that have the same variable raised to the same power. For instance, $3x^2$ and $7x^2$ are like terms because they both have the variable $x$ raised to the power of 2. Similarly, $3x$ and $-9x$ are like terms as they both contain the variable $x$ raised to the power of 1. Constants, such as 5 and 8, are also like terms. The process of addition involves adding the coefficients of these like terms while keeping the variable and its exponent unchanged.

Step-by-Step Solution

To accurately add the polynomials $\left(3x^2 + 3x + 5\right)$ and $\left(7x^2 - 9x + 8\right)$, we follow these steps:

1. Identify Like Terms: We identify terms with the same variable and exponent.

   • $3x^2$ and $7x^2$ are like terms.
   • $3x$ and $-9x$ are like terms.
   • $5$ and $8$ are like terms.

2. Combine Like Terms: We add the coefficients of the like terms.

   • For the $x^2$ terms: $3x^2 + 7x^2 = (3 + 7)x^2 = 10x^2$
   • For the $x$ terms: $3x + (-9x) = (3 - 9)x = -6x$
   • For the constant terms: $5 + 8 = 13$

3. Write the Simplified Polynomial: We combine the results to form the simplified polynomial.

   • $10x^2 - 6x + 13$

Detailed Explanation

Let's break down the process further. When adding polynomials, the distributive property and the commutative property of addition come into play. The distributive property allows us to group like terms together, while the commutative property allows us to change the order of terms without affecting the sum. For example, when adding $3x^2$ and $7x^2$, we are essentially combining three instances of $x^2$ with seven instances of $x^2$, resulting in ten instances of $x^2$, or $10x^2$. Similarly, when adding $3x$ and $-9x$, we are adding a positive multiple of $x$ to a negative multiple of $x$. This results in a net negative multiple of $x$, which in this case is $-6x$. The constants are straightforward to add; 5 and 8 combine to give 13. Therefore, the correct simplified polynomial is $10x^2 - 6x + 13$.

Pinpointing Lucia's Error

Analyzing Lucia's Result

Lucia's answer was $10x^2 - 12x + 13$. Comparing this with the correct answer, $10x^2 - 6x + 13$, we can see that the $x^2$ term and the constant term are correct. The discrepancy lies in the $x$ term. Lucia obtained $-12x$ instead of $-6x$.

Identifying the Mistake

To determine Lucia's error, let's re-examine the step where the $x$ terms are combined: $3x + (-9x)$. The correct operation is $3 - 9 = -6$, so the correct term is $-6x$. Lucia obtained $-12x$, which suggests that she might have incorrectly combined the coefficients. A possible error could be that she subtracted 9 from -3 instead of subtracting 9 from 3. Another possibility is that she might have added the absolute values of the coefficients (3 + 9 = 12) and kept the negative sign, leading to -12x. Either way, the mistake is in the arithmetic of combining the coefficients of the $x$ terms.

Possible Scenarios

1. Incorrect Subtraction: Lucia might have subtracted 9 from -3, thinking $3x + (-9x)$ as $-3x - 9x$, leading to $-12x$.
2. Adding Absolute Values: She might have added the absolute values of the coefficients, $3 + 9 = 12$, and retained the negative sign from $-9x$, resulting in $-12x$.
3. Misunderstanding Integer Arithmetic: A misunderstanding of how to add integers with different signs could also lead to this error. The rules of integer arithmetic dictate that when adding a positive number to a negative number, we subtract the smaller absolute value from the larger absolute value and keep the sign of the number with the larger absolute value. In this case, $|-9| > |3|$, so the result should have the same sign as $-9$, which is negative. The difference in absolute values is $|-9| - |3| = 9 - 3 = 6$, so the correct result is $-6x$.

Error Classification

Based on the analysis, Lucia's error is not in finding the difference instead of the sum, as the $x^2$ and constant terms were correctly added. The error lies specifically in combining the coefficients of the $x$ terms. This is an arithmetic error within the addition process, rather than a conceptual error of performing the wrong operation. Therefore, options A and B are incorrect. The correct assessment is that Lucia made an error in the arithmetic when combining the coefficients of the $x$ terms.

Rectifying the Error and Reinforcing Concepts

Correcting Lucia's Mistake

To correct Lucia's mistake, it's essential to revisit the steps of adding like terms, particularly those with different signs. The critical step is to accurately combine $3x$ and $-9x$. To do this, we remember that adding a negative number is the same as subtracting its positive counterpart. Therefore, $3x + (-9x)$ is equivalent to $3x - 9x$. The operation then becomes a subtraction of the coefficients: $3 - 9$. This results in $-6$, so the correct combined term is $-6x$.

Teaching Strategies

1. Visual Aids: Use visual aids, such as number lines, to illustrate the addition of integers with different signs. This can help Lucia visualize the movement along the number line when adding a positive and a negative number.
2. Real-World Examples: Provide real-world examples to make the concept more relatable. For example, consider temperatures: If the temperature is 3 degrees and it drops 9 degrees, what is the final temperature? This translates to $3 - 9 = -6$ degrees.
3. Practice Problems: Offer a variety of practice problems that involve adding and subtracting integers. This helps solidify the rules of integer arithmetic. Start with simple examples and gradually increase the complexity.
4. Step-by-Step Guidance: Encourage Lucia to break down each step and show her work. This makes it easier to identify where errors occur. For instance, she can write down $3x + (-9x)$, then rewrite it as $3x - 9x$, and finally calculate the result as $-6x$.
5. Review Integer Arithmetic: If the error stems from a misunderstanding of integer arithmetic, dedicate time to review the rules of adding, subtracting, multiplying, and dividing integers.

Conclusion

In summary, Lucia's error occurred in the process of combining the $x$ terms in the polynomial addition. She incorrectly calculated $3x + (-9x)$ as $-12x$ instead of $-6x$. This error likely stems from a mistake in integer arithmetic or a misapplication of the rules of addition and subtraction. By revisiting the steps of polynomial addition, focusing on integer arithmetic, and using visual aids and real-world examples, Lucia can correct her mistake and reinforce her understanding of algebraic principles. This detailed analysis not only pinpoints the error but also provides a comprehensive guide to prevent similar mistakes in the future, ensuring a solid foundation in polynomial manipulation.