Holly's Math Error: Unpacking Polynomial Subtraction

Hey math enthusiasts! Let's dive into a problem where our friend Holly seems to have stumbled a bit while working with polynomials. The original problem stated that Holly found that (11m - 13n + 6mn) - (10m - 7n + 3mn) = m - 20n + 9mn. But, did she actually get it right? Nope! We need to figure out what went wrong. The options given are:

A. She only used the additive inverse of 10m when combining like terms. B. She added the polynomials instead of subtracting. C. She only used the additive inverse of 7n when combining like terms.

Let's break down the polynomial subtraction step-by-step to understand the errors Holly could have made and identify the right answer.

Understanding Polynomial Subtraction: The Basics

Alright, before we start pointing fingers at Holly, let's quickly recap how polynomial subtraction should work. When subtracting polynomials, it's super important to remember that you're essentially distributing a negative sign across all the terms in the second polynomial. This changes the sign of each term within the parentheses being subtracted. It's like saying, "Hey, everything inside those parentheses? You're switching sides!" The general idea is to change the sign of each term in the polynomial being subtracted and then combine all the like terms. Like terms are terms that have the same variables raised to the same powers. So, for example, 3x and 7x are like terms, while 3x and 3x² are not. Remember, guys, the key concept here is distributing the negative sign. It's the most common area where mistakes occur. Let's see how this works with our specific problem.

So, when we have (11m - 13n + 6mn) - (10m - 7n + 3mn), the first step is to distribute that negative sign. It changes the equation to:

11m - 13n + 6mn - 10m + 7n - 3mn

Now, the next step is grouping like terms. Combine the 'm' terms, the 'n' terms, and the 'mn' terms separately. Combining the 'm' terms we get 11m - 10m = m. Combining the 'n' terms we get -13n + 7n = -6n. Finally, combining the 'mn' terms we get 6mn - 3mn = 3mn. Thus, the correct answer is m - 6n + 3mn. But Holly's result was m - 20n + 9mn. That is where something went wrong!

Analyzing Holly's Mistakes: Unraveling the Options

Let's go through the answer choices to see what went wrong. The goal is to figure out which statement correctly identifies the error Holly made when doing this subtraction. We already know the correct answer. Now, we are ready to find where Holly made an error.

A. She only used the additive inverse of 10m when combining like terms.

Let's see. If Holly only used the additive inverse of 10m, she would have correctly handled the m term. She would have computed 11m - 10m, which equals m. So, this option doesn't completely explain the errors in Holly's final answer. The provided result includes an incorrect coefficient in front of 'n' term and the 'mn' term. This suggests there were more than one mistakes. So, this option might be part of the mistake, but not the only mistake.

B. She added the polynomials instead of subtracting.

This is a likely one! If Holly added the polynomials instead of subtracting, she would have missed distributing the negative sign, which changes the sign of each term in the second polynomial. If we add the polynomials (11m - 13n + 6mn) and (10m - 7n + 3mn), we get 11m - 13n + 6mn + 10m - 7n + 3mn. Now combine like terms: (11m + 10m) + (-13n - 7n) + (6mn + 3mn) = 21m - 20n + 9mn. The answer produced by Holly (m - 20n + 9mn) has a few similarities. One of the terms is correct. It is possible that Holly added instead of subtracting, but also missed the sign change for the variable m. She got the sign correct for the variable 'n', which does not make sense. Therefore, although there is a match in one of the terms, the rest does not match, so we cannot completely accept this option.

C. She only used the additive inverse of 7n when combining like terms.

This option implies Holly only changed the sign of the '7n' term. As a result, she would have incorrectly handled the -13n term. If she only did this, the equation would have been 11m - 13n + 6mn - 10m + 7n - 3mn. Then, when you combine like terms you would get m - 6n + 3mn. The result does not match Holly's. Her answer has a different coefficient in front of the 'n' and 'mn' terms. Therefore, this option cannot be completely correct either.

Identifying Holly's Error: The Verdict

Let's look at Holly's incorrect answer again: m - 20n + 9mn. After distributing the negative sign, the correct calculation is: 11m - 13n + 6mn - 10m + 7n - 3mn. If Holly incorrectly added the polynomials, she would get: 11m - 13n + 6mn + 10m - 7n + 3mn, which simplifies to 21m - 20n + 9mn. Although the 'n' and 'mn' terms match the answer from Holly, the 'm' term does not match. Thus, the error cannot be completely due to adding polynomials. Another mistake must exist. Therefore, option A and C are incorrect. Therefore, the only logical choice should be B, with some additional errors.

The most likely scenario is Holly might have had a combination of errors: She potentially tried to add at first. After that, she might have missed the sign change of the 'm' term. Let's say she added the equations instead of subtracting: 11m - 13n + 6mn + 10m - 7n + 3mn = 21m - 20n + 9mn. From this result, she might have realized something was off, and decided to subtract. But she only subtracted the 10m term. That is 11m - 10m = m. Thus, Holly's actual answer would be m - 20n + 9mn. This will completely explain all the errors that were made. Therefore, although option B is not 100% accurate, it is closest to the truth.

Conclusion: Learning from Mistakes

So, what's the takeaway from Holly's experience? Always double-check your work, guys! Especially when dealing with polynomial subtraction, be extra careful with those negative signs. Distribute them correctly, group the like terms, and you'll be golden. Math is all about practice and learning from your mistakes. Keep up the great work, and you will eventually get the hang of it!