Question
Two classmates are working on multiplying the binomials (2x - 5)(x + 12) Their work is below: Joey Erin (2x - 5)(x + 12); (2x - 5)(x + 12); (2x)(x) + (2x)(12) + (- 5)(x) + (- 5)(12); (2x)(x) + (2x)(12) + (- 5)(x) + (- 5)(12); 2x + 24x - 5x - 602x ^ 2 + 24x - 5x - 60; 21x - 60; 2x ^ 2 + 19x - 60 Which classmate completed the problem correctly ? (1 point ) Explain what mistake the other classmate made (2 points) (3 points)
Answers
Answered by
GPT-4o mini
Let's analyze the work of both Joey and Erin to see who completed the multiplication of the binomials correctly.
### Joey's work:
1. Started with \((2x - 5)(x + 12)\) - correct.
2. Expanded to \((2x - 5)(x + 12)\) - correct.
3. Multiplied using the distributive property:
- \((2x)(x) + (2x)(12) + (-5)(x) + (-5)(12)\) - correct.
4. Calculated:
- \((2x)(x) = 2x^2\)
- \((2x)(12) = 24x\)
- \((-5)(x) = -5x\)
- \((-5)(12) = -60\)
5. Combined terms correctly:
- This turns into \(2x^2 + 24x - 5x - 60\), which reduces to \(2x^2 + 19x - 60\) - correct.
### Erin's work:
1. Started with \((2x - 5)(x + 12)\) - correct.
2. Expanded to \((2x - 5)(x + 12)\) - correct.
3. Multiplied using the distributive property:
- \((2x)(x) + (2x)(12) + (-5)(x) + (-5)(12)\) - correct.
4. Calculated:
- \((2x)(x) = 2x^2\)
- \((2x)(12) = 24x\)
- \((-5)(x) = -5x\)
- \((-5)(12) = -60\)
5. Combined terms:
- This is where Erin made a mistake. She wrote down \(2x + 24x - 5x - 60\) instead, which is incorrect.
### Conclusion:
**Joey is correct.**
### Mistake Analysis:
Erin mistakenly wrote \(2x + 24x - 5x - 60\) instead of \(2x^2 + 24x - 5x - 60\) in her combination of terms. This error resulted from incorrectly omitting the \(2x^2\) term and directly starting with the linear terms, leading to an incorrect polynomial \(21x - 60\). The correct simplification involves keeping the \(2x^2\) term, and thus the correct final polynomial should be \(2x^2 + 19x - 60\).
### Joey's work:
1. Started with \((2x - 5)(x + 12)\) - correct.
2. Expanded to \((2x - 5)(x + 12)\) - correct.
3. Multiplied using the distributive property:
- \((2x)(x) + (2x)(12) + (-5)(x) + (-5)(12)\) - correct.
4. Calculated:
- \((2x)(x) = 2x^2\)
- \((2x)(12) = 24x\)
- \((-5)(x) = -5x\)
- \((-5)(12) = -60\)
5. Combined terms correctly:
- This turns into \(2x^2 + 24x - 5x - 60\), which reduces to \(2x^2 + 19x - 60\) - correct.
### Erin's work:
1. Started with \((2x - 5)(x + 12)\) - correct.
2. Expanded to \((2x - 5)(x + 12)\) - correct.
3. Multiplied using the distributive property:
- \((2x)(x) + (2x)(12) + (-5)(x) + (-5)(12)\) - correct.
4. Calculated:
- \((2x)(x) = 2x^2\)
- \((2x)(12) = 24x\)
- \((-5)(x) = -5x\)
- \((-5)(12) = -60\)
5. Combined terms:
- This is where Erin made a mistake. She wrote down \(2x + 24x - 5x - 60\) instead, which is incorrect.
### Conclusion:
**Joey is correct.**
### Mistake Analysis:
Erin mistakenly wrote \(2x + 24x - 5x - 60\) instead of \(2x^2 + 24x - 5x - 60\) in her combination of terms. This error resulted from incorrectly omitting the \(2x^2\) term and directly starting with the linear terms, leading to an incorrect polynomial \(21x - 60\). The correct simplification involves keeping the \(2x^2\) term, and thus the correct final polynomial should be \(2x^2 + 19x - 60\).