Question
Order the steps from first to last that would be used to factor:
3z2-13-10
(7 points)
Determine if there is a greatest common factor amongst all the terms.
Find the two numbers that give a product of -30 and a sum of-13.
Replace the middle term with-152 and 2
Factor by grouping: Take the greatest common factor out of the first two terms and the greatest common factor out of the last two terms to get 3r (5)+2(x-5)
Multiply 3x-10
Factor the common factor orice more to get your final answer.
Final answer (-5) (3+2)
3z2-13-10
(7 points)
Determine if there is a greatest common factor amongst all the terms.
Find the two numbers that give a product of -30 and a sum of-13.
Replace the middle term with-152 and 2
Factor by grouping: Take the greatest common factor out of the first two terms and the greatest common factor out of the last two terms to get 3r (5)+2(x-5)
Multiply 3x-10
Factor the common factor orice more to get your final answer.
Final answer (-5) (3+2)
Answers
GPT-4o mini
To correctly factor the expression \( 3z^2 - 13z - 10 \), we can follow these ordered steps:
1. **Determine if there is a greatest common factor amongst all the terms.**
- Check the coefficients for any common factors. In this case, the coefficients are 3, -13, and -10. There is no common factor among these coefficients.
2. **Find the two numbers that give a product of -30 and a sum of -13.**
- We need to factor the expression \( 3z^2 - 13z - 10 \). The product of the coefficients of \( z^2 \) and the constant term is \( 3 \times -10 = -30 \). We need to find two numbers that multiply to -30 and add to -13. The numbers -15 and 2 work here (-15 * 2 = -30 and -15 + 2 = -13).
3. **Replace the middle term with the two numbers found -15 and 2.**
- Rewrite the expression: \( 3z^2 - 15z + 2z - 10 \).
4. **Factor by grouping: Take the greatest common factor out of the first two terms and the greatest common factor out of the last two terms.**
- Group the terms: \( (3z^2 - 15z) + (2z - 10) \).
- Factor out \( 3z \) from the first group and \( 2 \) from the second group:
- \( 3z(z - 5) + 2(z - 5) \).
5. **Factor out the common factor (z - 5) once more to get your final answer.**
- Now, we can factor out \( (z - 5) \):
- \( (z - 5)(3z + 2) \).
6. **Final answer: \( (z - 5)(3z + 2) \)**.
So the ordered steps to factor \( 3z^2 - 13z - 10 \) are:
1. Determine if there is a greatest common factor amongst all the terms.
2. Find the two numbers that give a product of -30 and a sum of -13.
3. Replace the middle term with -15 and 2.
4. Factor by grouping.
5. Factor out the common factor once more to get your final answer.
6. Final answer: \( (z - 5)(3z + 2) \).
(Note: There were typographical errors in the original text, and some parts of the original procedure's text do not strictly correspond to the expression being factored, such as “Multiply 3x-10.” The correct expression is indeed \( 3z^2 - 13z - 10 \) not \( 3x - 10 \)).
1. **Determine if there is a greatest common factor amongst all the terms.**
- Check the coefficients for any common factors. In this case, the coefficients are 3, -13, and -10. There is no common factor among these coefficients.
2. **Find the two numbers that give a product of -30 and a sum of -13.**
- We need to factor the expression \( 3z^2 - 13z - 10 \). The product of the coefficients of \( z^2 \) and the constant term is \( 3 \times -10 = -30 \). We need to find two numbers that multiply to -30 and add to -13. The numbers -15 and 2 work here (-15 * 2 = -30 and -15 + 2 = -13).
3. **Replace the middle term with the two numbers found -15 and 2.**
- Rewrite the expression: \( 3z^2 - 15z + 2z - 10 \).
4. **Factor by grouping: Take the greatest common factor out of the first two terms and the greatest common factor out of the last two terms.**
- Group the terms: \( (3z^2 - 15z) + (2z - 10) \).
- Factor out \( 3z \) from the first group and \( 2 \) from the second group:
- \( 3z(z - 5) + 2(z - 5) \).
5. **Factor out the common factor (z - 5) once more to get your final answer.**
- Now, we can factor out \( (z - 5) \):
- \( (z - 5)(3z + 2) \).
6. **Final answer: \( (z - 5)(3z + 2) \)**.
So the ordered steps to factor \( 3z^2 - 13z - 10 \) are:
1. Determine if there is a greatest common factor amongst all the terms.
2. Find the two numbers that give a product of -30 and a sum of -13.
3. Replace the middle term with -15 and 2.
4. Factor by grouping.
5. Factor out the common factor once more to get your final answer.
6. Final answer: \( (z - 5)(3z + 2) \).
(Note: There were typographical errors in the original text, and some parts of the original procedure's text do not strictly correspond to the expression being factored, such as “Multiply 3x-10.” The correct expression is indeed \( 3z^2 - 13z - 10 \) not \( 3x - 10 \)).