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Answer :
Certainly! Let’s go through the steps in a detailed manner to find the correct subtraction and analyze the error made by Martina.
First, we need to determine the correct subtraction of the two polynomials. Let's denote the polynomials as follows:
[tex]\( \text{Polynomial 1: } 14c + 31d - 20 \)[/tex]
[tex]\( \text{Polynomial 2: } 17c - 24d + 9 \)[/tex]
### Correct Difference Calculation
To subtract the second polynomial from the first, we need to subtract each term of the second polynomial from the corresponding term of the first polynomial:
[tex]\[
(14c + 31d - 20) - (17c - 24d + 9)
\][/tex]
This means we perform the following subtraction term by term:
1. Subtract the [tex]\( c \)[/tex]-terms:
[tex]\[ 14c - 17c = -3c \][/tex]
2. Subtract the [tex]\( d \)[/tex]-terms:
[tex]\[ 31d - (-24d) = 31d + 24d = 55d \][/tex]
3. Subtract the constant terms:
[tex]\[ -20 - 9 = -29 \][/tex]
Combining these results, we get:
[tex]\[ -3c + 55d - 29 \][/tex]
This is the correct difference.
### Martina's Incorrect Subtraction
Martina mistakenly subtracted and obtained [tex]\( 9et - 3c + 7d - 29 \)[/tex].
### Identifying the Error
Let's compare the correct result with Martina's result. The correct result is:
[tex]\[ -3c + 55d - 29 \][/tex]
Martina's result is:
[tex]\[ 9et - 3c + 7d - 29 \][/tex]
### Analyze the Error
Martina’s result matches the correct result in the [tex]\(c\)[/tex]-terms and the constant term:
- The [tex]\(c\)[/tex]-term: Both results have [tex]\(-3c\)[/tex]
- The constant term: Both results have [tex]\(-29\)[/tex]
However, Martina made an error in the [tex]\(d\)[/tex]-term and introduced an extra term [tex]\(9et\)[/tex]:
- The [tex]\(d\)[/tex]-term in the correct result is [tex]\(55d\)[/tex], but Martina has [tex]\(7d\)[/tex]. To make the comparison:
[tex]\[ 55d - 7d = 48d \][/tex]
So, she missed [tex]\(48d\)[/tex] in her calculation.
Additionally, she introduced an erroneous term [tex]\(9et\)[/tex] which should not exist in the subtraction.
### Conclusion
The error made by Martina includes:
1. Missed adding [tex]\(48d\)[/tex] in her result.
2. Erroneously introduced an extra term [tex]\(9et\)[/tex].
So, the missing error to correct her subtraction will be [tex]\(48d\)[/tex] and the introduced incorrect term is [tex]\(9et\)[/tex]. Therefore, the total error Martina made can be expressed as:
[tex]\[ 55d - 7d + 9et = 48d + 9et \][/tex]
First, we need to determine the correct subtraction of the two polynomials. Let's denote the polynomials as follows:
[tex]\( \text{Polynomial 1: } 14c + 31d - 20 \)[/tex]
[tex]\( \text{Polynomial 2: } 17c - 24d + 9 \)[/tex]
### Correct Difference Calculation
To subtract the second polynomial from the first, we need to subtract each term of the second polynomial from the corresponding term of the first polynomial:
[tex]\[
(14c + 31d - 20) - (17c - 24d + 9)
\][/tex]
This means we perform the following subtraction term by term:
1. Subtract the [tex]\( c \)[/tex]-terms:
[tex]\[ 14c - 17c = -3c \][/tex]
2. Subtract the [tex]\( d \)[/tex]-terms:
[tex]\[ 31d - (-24d) = 31d + 24d = 55d \][/tex]
3. Subtract the constant terms:
[tex]\[ -20 - 9 = -29 \][/tex]
Combining these results, we get:
[tex]\[ -3c + 55d - 29 \][/tex]
This is the correct difference.
### Martina's Incorrect Subtraction
Martina mistakenly subtracted and obtained [tex]\( 9et - 3c + 7d - 29 \)[/tex].
### Identifying the Error
Let's compare the correct result with Martina's result. The correct result is:
[tex]\[ -3c + 55d - 29 \][/tex]
Martina's result is:
[tex]\[ 9et - 3c + 7d - 29 \][/tex]
### Analyze the Error
Martina’s result matches the correct result in the [tex]\(c\)[/tex]-terms and the constant term:
- The [tex]\(c\)[/tex]-term: Both results have [tex]\(-3c\)[/tex]
- The constant term: Both results have [tex]\(-29\)[/tex]
However, Martina made an error in the [tex]\(d\)[/tex]-term and introduced an extra term [tex]\(9et\)[/tex]:
- The [tex]\(d\)[/tex]-term in the correct result is [tex]\(55d\)[/tex], but Martina has [tex]\(7d\)[/tex]. To make the comparison:
[tex]\[ 55d - 7d = 48d \][/tex]
So, she missed [tex]\(48d\)[/tex] in her calculation.
Additionally, she introduced an erroneous term [tex]\(9et\)[/tex] which should not exist in the subtraction.
### Conclusion
The error made by Martina includes:
1. Missed adding [tex]\(48d\)[/tex] in her result.
2. Erroneously introduced an extra term [tex]\(9et\)[/tex].
So, the missing error to correct her subtraction will be [tex]\(48d\)[/tex] and the introduced incorrect term is [tex]\(9et\)[/tex]. Therefore, the total error Martina made can be expressed as:
[tex]\[ 55d - 7d + 9et = 48d + 9et \][/tex]
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