Question
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Grace, Chelsea, and Roan are simplifying the same polynomial expression. Which student’s work is correct and why?
Grace Chelsea Roan
3(2 − x) − 2(6x − 8) 3(2 − x) − 2(6x − 8) 3(2 − x) − 2(6x − 8)
= 6 − 3x − 12x + 16 = 6 − 3x − 12x – 16 = 6 − 3x + 12x – 16
= -3x − 12x + 16 + 6 = -3x − 12x − 16 + 6 = -3x + 12x − 16 + 6
= -15x + 22 = -15x − 10 = 9x − 10
Grace, Chelsea, and Roan are simplifying the same polynomial expression. Which student’s work is correct and why?
Grace Chelsea Roan
3(2 − x) − 2(6x − 8) 3(2 − x) − 2(6x − 8) 3(2 − x) − 2(6x − 8)
= 6 − 3x − 12x + 16 = 6 − 3x − 12x – 16 = 6 − 3x + 12x – 16
= -3x − 12x + 16 + 6 = -3x − 12x − 16 + 6 = -3x + 12x − 16 + 6
= -15x + 22 = -15x − 10 = 9x − 10
Answers
GPT-4o mini
Let's analyze each student's work step by step to determine whose simplification of the polynomial expression is correct.
Starting with the original expression that all three students used:
\[ 3(2 − x) − 2(6x − 8) \]
1. **Distributing the terms**:
- For Grace:
\[ 3(2) - 3(x) - 2(6x) + 2(8) \]
\[ = 6 - 3x - 12x + 16 \]
- For Chelsea:
\[ 3(2) - 3(x) - 2(6x) + 2(8) \]
\[ = 6 - 3x - 12x - 16 \]
- For Roan:
\[ 3(2) - 3(x) - 2(6x) + 2(8) \]
\[ = 6 - 3x + 12x - 16 \]
2. **Combining like terms**:
- Grace's terms:
\[ 6 + 16 - 3x - 12x = 22 - 15x \rightarrow -15x + 22 \]
- Chelsea's terms:
\[ 6 - 16 - 3x - 12x = -10 - 15x \rightarrow -15x - 10 \]
- Roan's terms:
\[ 6 - 16 - 3x + 12x = -10 + 9x \rightarrow 9x - 10 \]
Now, we check the combination of like terms step:
For Grace’s calculation:
- Combine numerical constants: \( 6 + 16 = 22 \)
- Combine \( x \) terms: \( -3x - 12x = -15x \)
Thus, Grace’s final simplified form is \(-15x + 22\).
For Chelsea’s calculation:
- Combine numerical constants: \( 6 - 16 = -10 \)
- Combine \( x \) terms: \( -3x - 12x = -15x \)
So, Chelsea's final form is \(-15x - 10\).
For Roan’s calculation:
- Combine numerical constants: \( 6 - 16 = -10 \)
- Combine \( x \) terms: \( -3x + 12x = 9x \)
His final form is \(9x - 10\).
3. **Final comparison**:
- Grace: \(-15x + 22\)
- Chelsea: \(-15x - 10\)
- Roan: \(9x - 10\)
The correct simplification of the polynomial expression results in \(-15x + 22\). Thus, **Grace’s work is correct**. Chelsea and Roan made mistakes in their simplifications either by incorrectly distributing the terms or combining like terms. Grace maintained the correct distribution and combination throughout her work.
Starting with the original expression that all three students used:
\[ 3(2 − x) − 2(6x − 8) \]
1. **Distributing the terms**:
- For Grace:
\[ 3(2) - 3(x) - 2(6x) + 2(8) \]
\[ = 6 - 3x - 12x + 16 \]
- For Chelsea:
\[ 3(2) - 3(x) - 2(6x) + 2(8) \]
\[ = 6 - 3x - 12x - 16 \]
- For Roan:
\[ 3(2) - 3(x) - 2(6x) + 2(8) \]
\[ = 6 - 3x + 12x - 16 \]
2. **Combining like terms**:
- Grace's terms:
\[ 6 + 16 - 3x - 12x = 22 - 15x \rightarrow -15x + 22 \]
- Chelsea's terms:
\[ 6 - 16 - 3x - 12x = -10 - 15x \rightarrow -15x - 10 \]
- Roan's terms:
\[ 6 - 16 - 3x + 12x = -10 + 9x \rightarrow 9x - 10 \]
Now, we check the combination of like terms step:
For Grace’s calculation:
- Combine numerical constants: \( 6 + 16 = 22 \)
- Combine \( x \) terms: \( -3x - 12x = -15x \)
Thus, Grace’s final simplified form is \(-15x + 22\).
For Chelsea’s calculation:
- Combine numerical constants: \( 6 - 16 = -10 \)
- Combine \( x \) terms: \( -3x - 12x = -15x \)
So, Chelsea's final form is \(-15x - 10\).
For Roan’s calculation:
- Combine numerical constants: \( 6 - 16 = -10 \)
- Combine \( x \) terms: \( -3x + 12x = 9x \)
His final form is \(9x - 10\).
3. **Final comparison**:
- Grace: \(-15x + 22\)
- Chelsea: \(-15x - 10\)
- Roan: \(9x - 10\)
The correct simplification of the polynomial expression results in \(-15x + 22\). Thus, **Grace’s work is correct**. Chelsea and Roan made mistakes in their simplifications either by incorrectly distributing the terms or combining like terms. Grace maintained the correct distribution and combination throughout her work.