Westonci.ca makes finding answers easy, with a community of experts ready to provide you with the information you seek. Explore in-depth answers to your questions from a knowledgeable community of experts across different fields. Get precise and detailed answers to your questions from a knowledgeable community of experts on our Q&A platform.
Sagot :
Certainly! Let's analyze the factoring approaches of both Lucas and Erick step by step.
### Lucas' Method:
Lucas groups the polynomial as:
[tex]\[ (12x^3 + 8x) + (-6x^2 - 4) \][/tex]
1. First Group: [tex]\( 12x^3 + 8x \)[/tex]
- Identify common factors:
[tex]\[ 12x^3 + 8x = 4x(3x^2 + 2) \][/tex]
2. Second Group: [tex]\( -6x^2 - 4 \)[/tex]
- Identify common factors:
[tex]\[ -6x^2 - 4 = -2(3x^2 + 2) \][/tex]
3. Combine Factored Terms:
- Combine the factored forms within each group:
[tex]\[ 4x(3x^2 + 2) + (-2)(3x^2 + 2) \][/tex]
- Factor out the common binomial term [tex]\( (3x^2 + 2) \)[/tex]:
[tex]\[ (3x^2 + 2)(4x - 2) \][/tex]
### Erick's Method:
Erick groups the polynomial as:
[tex]\[ (12x^3 - 6x^2) + (8x - 4) \][/tex]
1. First Group: [tex]\( 12x^3 - 6x^2 \)[/tex]
- Identify common factors:
[tex]\[ 12x^3 - 6x^2 = 6x^2(2x - 1) \][/tex]
2. Second Group: [tex]\( 8x - 4 \)[/tex]
- Identify common factors:
[tex]\[ 8x - 4 = 4(2x - 1) \][/tex]
3. Combine Factored Terms:
- Combine the factored forms within each group:
[tex]\[ 6x^2(2x - 1) + 4(2x - 1) \][/tex]
- Factor out the common binomial term [tex]\( (2x - 1) \)[/tex]:
[tex]\[ (2x - 1)(6x^2 + 4) \][/tex]
### Verification:
1. Lucas' Factored Form: [tex]\((3x^2 + 2)(4x - 2)\)[/tex]
- Expand and verify if it matches the original polynomial:
[tex]\[ (3x^2 + 2)(4x - 2) = 3x^2 \cdot 4x + 3x^2 \cdot -2 + 2 \cdot 4x + 2 \cdot -2 \][/tex]
[tex]\[ = 12x^3 -6x^2 + 8x - 4 \][/tex]
- The factorization is correct as it gives the original polynomial.
2. Erick's Factored Form: [tex]\((2x - 1)(6x^2 + 4)\)[/tex]
- Expand and verify if it matches the original polynomial:
[tex]\[ (2x - 1)(6x^2 + 4) = 2x(6x^2) + 2x(4) - 1(6x^2) - 1(4) \][/tex]
[tex]\[ = 12x^3 + 8x - 6x^2 - 4 \][/tex]
- The factorization is also correct as it gives the original polynomial.
### Conclusion:
Both Lucas and Erick have correctly grouped the terms of the polynomial [tex]\( 12x^3 - 6x^2 + 8x - 4\)[/tex] and factored it properly. Therefore, both students are correct in their approaches.
### Lucas' Method:
Lucas groups the polynomial as:
[tex]\[ (12x^3 + 8x) + (-6x^2 - 4) \][/tex]
1. First Group: [tex]\( 12x^3 + 8x \)[/tex]
- Identify common factors:
[tex]\[ 12x^3 + 8x = 4x(3x^2 + 2) \][/tex]
2. Second Group: [tex]\( -6x^2 - 4 \)[/tex]
- Identify common factors:
[tex]\[ -6x^2 - 4 = -2(3x^2 + 2) \][/tex]
3. Combine Factored Terms:
- Combine the factored forms within each group:
[tex]\[ 4x(3x^2 + 2) + (-2)(3x^2 + 2) \][/tex]
- Factor out the common binomial term [tex]\( (3x^2 + 2) \)[/tex]:
[tex]\[ (3x^2 + 2)(4x - 2) \][/tex]
### Erick's Method:
Erick groups the polynomial as:
[tex]\[ (12x^3 - 6x^2) + (8x - 4) \][/tex]
1. First Group: [tex]\( 12x^3 - 6x^2 \)[/tex]
- Identify common factors:
[tex]\[ 12x^3 - 6x^2 = 6x^2(2x - 1) \][/tex]
2. Second Group: [tex]\( 8x - 4 \)[/tex]
- Identify common factors:
[tex]\[ 8x - 4 = 4(2x - 1) \][/tex]
3. Combine Factored Terms:
- Combine the factored forms within each group:
[tex]\[ 6x^2(2x - 1) + 4(2x - 1) \][/tex]
- Factor out the common binomial term [tex]\( (2x - 1) \)[/tex]:
[tex]\[ (2x - 1)(6x^2 + 4) \][/tex]
### Verification:
1. Lucas' Factored Form: [tex]\((3x^2 + 2)(4x - 2)\)[/tex]
- Expand and verify if it matches the original polynomial:
[tex]\[ (3x^2 + 2)(4x - 2) = 3x^2 \cdot 4x + 3x^2 \cdot -2 + 2 \cdot 4x + 2 \cdot -2 \][/tex]
[tex]\[ = 12x^3 -6x^2 + 8x - 4 \][/tex]
- The factorization is correct as it gives the original polynomial.
2. Erick's Factored Form: [tex]\((2x - 1)(6x^2 + 4)\)[/tex]
- Expand and verify if it matches the original polynomial:
[tex]\[ (2x - 1)(6x^2 + 4) = 2x(6x^2) + 2x(4) - 1(6x^2) - 1(4) \][/tex]
[tex]\[ = 12x^3 + 8x - 6x^2 - 4 \][/tex]
- The factorization is also correct as it gives the original polynomial.
### Conclusion:
Both Lucas and Erick have correctly grouped the terms of the polynomial [tex]\( 12x^3 - 6x^2 + 8x - 4\)[/tex] and factored it properly. Therefore, both students are correct in their approaches.
We appreciate your time. Please revisit us for more reliable answers to any questions you may have. Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. Your questions are important to us at Westonci.ca. Visit again for expert answers and reliable information.
