Holly's Mistake In Polynomial Subtraction Analyzing The Error

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In this article, we delve into a common algebraic error made by a student named Holly while simplifying a polynomial expression. Holly attempted to simplify the expression $(11m - 13n + 6mn) - (10m - 7n + 3mn)$ and arrived at the result $m - 20n + 9mn$. Our goal is to identify the specific error Holly made during the subtraction process. This exercise is crucial for understanding the nuances of polynomial arithmetic and avoiding similar mistakes in the future.

Understanding the Problem

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The problem presented involves subtracting one polynomial from another. Polynomials are algebraic expressions consisting of variables and coefficients, combined using addition, subtraction, and multiplication. The given expression, $(11m - 13n + 6mn) - (10m - 7n + 3mn)$, contains three terms in each polynomial: terms with the variable 'm', terms with the variable 'n', and terms with the product 'mn'.

To correctly subtract polynomials, it's essential to distribute the negative sign to each term within the second polynomial and then combine like terms. Like terms are terms that have the same variables raised to the same powers. For instance, $11m$ and $10m$ are like terms, $-13n$ and $-7n$ are like terms, and $6mn$ and $3mn$ are like terms. The subtraction process involves combining the coefficients of these like terms after distributing the negative sign.

Holly's error likely occurred during this distribution or combination process. By carefully examining each step, we can pinpoint where the mistake was made. It’s also important to remember the fundamental rules of algebra, such as the additive inverse property, which states that subtracting a number is the same as adding its negative. This property is crucial when dealing with polynomial subtraction.

Step-by-Step Solution

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To accurately identify Holly's mistake, let's meticulously solve the problem step-by-step. This will provide a clear comparison point against Holly's incorrect answer and illuminate the specific error she committed.

1. Distribute the Negative Sign: The first crucial step in subtracting polynomials is to distribute the negative sign (the minus sign) in front of the second polynomial to each term inside the parentheses. This means we change the sign of each term in the second polynomial:

   $(11m - 13n + 6mn) - (10m - 7n + 3mn) = 11m - 13n + 6mn - 10m + 7n - 3mn$

   Notice how the signs of $10m$, $-7n$, and $3mn$ have been flipped to $-10m$, $+7n$, and $-3mn$, respectively. This is a direct application of the distributive property and the additive inverse concept.

2. Identify Like Terms: Next, we need to identify and group the like terms. As mentioned earlier, like terms have the same variable components. In this expression, we have:

   • 'm' terms: $11m$ and $-10m$
   • 'n' terms: $-13n$ and $+7n$
   • 'mn' terms: $6mn$ and $-3mn$

   Grouping these terms together helps in the next step of combining them efficiently and accurately.

3. Combine Like Terms: Now, we combine the coefficients of the like terms by adding or subtracting them as indicated:

   • Combine 'm' terms: $11m - 10m = (11 - 10)m = 1m = m$
   • Combine 'n' terms: $-13n + 7n = (-13 + 7)n = -6n$
   • Combine 'mn' terms: $6mn - 3mn = (6 - 3)mn = 3mn$

   This step is where careful arithmetic is crucial. Each pair of like terms is simplified by performing the necessary operation on their coefficients.

4. Write the Simplified Expression: Finally, we write the simplified expression by combining the results from the previous step:

   $m - 6n + 3mn$

   This is the correct simplified form of the given expression. Now, we can compare this with Holly's answer to see where she went wrong.

Identifying Holly's Error

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Holly's answer was $m - 20n + 9mn$. Comparing this to the correct answer, $m - 6n + 3mn$, we can clearly see discrepancies in the 'n' and 'mn' terms. The 'm' term is correct, indicating that Holly likely performed the subtraction for the 'm' terms accurately.

Let's break down the possible errors:

1. Error in the 'n' term: Holly obtained $-20n$ instead of the correct $-6n$. This suggests an error in combining the 'n' terms. The correct operation was $-13n + 7n = -6n$. It's possible that Holly added the coefficients instead of subtracting, or made a sign error during the addition.

2. Error in the 'mn' term: Holly obtained $9mn$ instead of the correct $3mn$. This indicates an error in combining the 'mn' terms. The correct operation was $6mn - 3mn = 3mn$. A likely error here is that Holly might have added the coefficients instead of subtracting them.

Based on this analysis, the most probable errors Holly made are:

• Incorrectly combining the 'n' terms: She likely performed an incorrect operation on the coefficients $-13$ and $+7$, possibly adding them instead of subtracting or making a sign mistake.
• Incorrectly combining the 'mn' terms: She likely added the coefficients $6$ and $3$ instead of subtracting them.

Therefore, the most comprehensive answer is that Holly made errors in both the 'n' and 'mn' terms by not correctly applying the subtraction operation and potentially mishandling the signs of the coefficients.

Analyzing the Answer Choices

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Now, let's consider the answer choices provided:

A. She only used the additive inverse of 10 m when combining like terms. B. She added the polynomials instead of subtracting. C. She only used the additive inverse

Let's analyze each choice in detail:

• Choice A: She only used the additive inverse of 10 m when combining like terms. This statement is partially correct. Holly did need to use the additive inverse of $10m$, which is $-10m$. However, this choice implies that the error was isolated to the 'm' term, which we know is not the case since Holly correctly combined the 'm' terms. The error lies in the 'n' and 'mn' terms, making this choice incomplete.

• Choice B: She added the polynomials instead of subtracting. This is a strong contender. If Holly added the polynomials instead of subtracting, she would have performed the operation $(11m - 13n + 6mn) + (10m - 7n + 3mn)$. Let's see what result this yields:

  $11m - 13n + 6mn + 10m - 7n + 3mn = (11m + 10m) + (-13n - 7n) + (6mn + 3mn) = 21m - 20n + 9mn$

  Notice that the 'n' and 'mn' terms match Holly's incorrect answer ($m - 20n + 9mn$). This strongly suggests that Holly's primary error was adding the polynomials instead of subtracting. The 'm' term doesn't match, but this could be a secondary error or a result of misremembering the initial expression.

• Choice C: She only used the additive inverse... This choice is incomplete as it doesn't provide a full statement, and we need a complete explanation of the error.

Based on this analysis, Choice B is the most likely correct answer. Holly's primary error was adding the polynomials instead of subtracting them, which explains the incorrect 'n' and 'mn' terms in her answer.

Conclusion

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In conclusion, by carefully analyzing Holly's answer and comparing it to the step-by-step solution, we identified that the most likely error Holly made was adding the polynomials instead of subtracting them (Choice B). This error explains the discrepancies in the 'n' and 'mn' terms of her answer. While there might have been secondary errors, such as sign mishandling, the primary error was the incorrect operation performed between the two polynomials.

This exercise highlights the importance of meticulously following the rules of algebra, especially when dealing with polynomial operations. Distributing the negative sign correctly and combining like terms accurately are crucial steps in simplifying algebraic expressions. By understanding these concepts and practicing regularly, students can avoid making similar mistakes and build a strong foundation in algebra.

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• Polynomial Subtraction Error
• Algebraic Simplification
• Combining Like Terms
• Distributive Property
• Additive Inverse
• Holly's Error Analysis
• Mathematics Education
• Polynomial Arithmetic
• Algebraic Expressions
• Math Mistakes
• Identifying Errors in Algebra
• Solving Polynomial Problems
• Step-by-Step Solution
• Common Algebra Mistakes
• Polynomial Operations

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