Answer :
To determine which products result in a sum or difference of cubes, let's analyze each product one by one according to the formulas for the sum and difference of cubes:
1. [tex]\((x-4)\left(x^2+4x-16\right)\)[/tex]
- Difference of Cubes Formula: [tex]\((a-b)(a^2+ab+b^2) = a^3 - b^3\)[/tex]
Identify [tex]\(a = x\)[/tex] and [tex]\(b = 4\)[/tex]:
[tex]\[ a^2 + ab + b^2 = x^2 + 4x + 16 \][/tex]
However, the given polynomial is [tex]\(x^2 + 4x - 16\)[/tex], which does not match [tex]\(x^2 + 4x + 16\)[/tex].
- Sum of Cubes Formula: [tex]\((a+b)(a^2-ab+b^2) = a^3 + b^3\)[/tex]
Identify [tex]\(a = x\)[/tex] and [tex]\(b = 4\)[/tex]:
[tex]\[ a^2 - ab + b^2 = x^2 - 4x + 16 \][/tex]
However, the given polynomial is [tex]\(x^2 + 4x - 16\)[/tex].
This product does not fit either formula.
2. [tex]\((x-1)\left(x^2-x+1\right)\)[/tex]
- Difference of Cubes Formula: [tex]\((a-b)(a^2+ab+b^2) = a^3 - b^3\)[/tex]
Identify [tex]\(a = x\)[/tex] and [tex]\(b = 1\)[/tex]:
[tex]\[ a^2 + ab + b^2 = x^2 + x + 1 \][/tex]
However, the given polynomial is [tex]\(x^2 - x + 1\)[/tex], which does not match [tex]\(x^2 + x + 1\)[/tex].
This product also does not fit the formula for difference of cubes.
3. [tex]\((x-1)\left(x^2+x+1\right)\)[/tex]
- Difference of Cubes Formula: [tex]\((a-b)(a^2+ab+b^2) = a^3 - b^3\)[/tex]
Identify [tex]\(a = x\)[/tex] and [tex]\(b = 1\)[/tex]:
[tex]\[ a^2 + ab + b^2 = x^2 + x + 1 \][/tex]
The given polynomial [tex]\(x^2+x+1\)[/tex] matches [tex]\(x^2 + ab + b^2\)[/tex].
This product fits the difference of cubes formula and results in [tex]\(x^3 - 1\)[/tex].
4. [tex]\((x+1)\left(x^2+x-1\right)\)[/tex]
- Sum of Cubes Formula: [tex]\((a+b)(a^2-ab+b^2) = a^3 + b^3\)[/tex]
Identify [tex]\(a = x\)[/tex] and [tex]\(b = 1\)[/tex]:
[tex]\[ a^2 - ab + b^2 = x^2 - x + 1 \][/tex]
However, the given polynomial is [tex]\(x^2 + x - 1\)[/tex], which does not match [tex]\(x^2 - x + 1\)[/tex].
This product does not fit either formula.
5. [tex]\((x+4)\left(x^2-4x+16\right)\)[/tex]
- Sum of Cubes Formula: [tex]\((a+b)(a^2-ab+b^2) = a^3 + b^3\)[/tex]
Identify [tex]\(a = x\)[/tex] and [tex]\(b = 4\)[/tex]:
[tex]\[ a^2 - ab + b^2 = x^2 - 4x + 16 \][/tex]
The given polynomial [tex]\(x^2-4x+16\)[/tex] matches [tex]\(a^2 - ab + b^2\)[/tex].
This product fits the sum of cubes formula and results in [tex]\(x^3 + 64\)[/tex].
6. [tex]\((x+4)\left(x^2+4x+16\right)\)[/tex]
- Sum of Cubes Formula: [tex]\((a+b)(a^2-ab+b^2) = a^3 + b^3\)[/tex]
Identify [tex]\(a = x\)[/tex] and [tex]\(b = 4\)[/tex]:
[tex]\[ a^2 - ab + b^2 = x^2 - 4x + 16 \][/tex]
However, the given polynomial is [tex]\(x^2 + 4x + 16\)[/tex], which does not match [tex]\(x^2 - 4x + 16\)[/tex].
This product does not fit either formula.
In conclusion, the products that result in a sum or difference of cubes are:
[tex]\[ (x-1)\left(x^2+x+1\right) \quad \text{and} \quad (x+4)\left(x^2-4x+16\right) \][/tex]
Thus, matching the results gives:
[tex]\[ [0, 0, 2, 0, 4, 0] \][/tex]
1. [tex]\((x-4)\left(x^2+4x-16\right)\)[/tex]
- Difference of Cubes Formula: [tex]\((a-b)(a^2+ab+b^2) = a^3 - b^3\)[/tex]
Identify [tex]\(a = x\)[/tex] and [tex]\(b = 4\)[/tex]:
[tex]\[ a^2 + ab + b^2 = x^2 + 4x + 16 \][/tex]
However, the given polynomial is [tex]\(x^2 + 4x - 16\)[/tex], which does not match [tex]\(x^2 + 4x + 16\)[/tex].
- Sum of Cubes Formula: [tex]\((a+b)(a^2-ab+b^2) = a^3 + b^3\)[/tex]
Identify [tex]\(a = x\)[/tex] and [tex]\(b = 4\)[/tex]:
[tex]\[ a^2 - ab + b^2 = x^2 - 4x + 16 \][/tex]
However, the given polynomial is [tex]\(x^2 + 4x - 16\)[/tex].
This product does not fit either formula.
2. [tex]\((x-1)\left(x^2-x+1\right)\)[/tex]
- Difference of Cubes Formula: [tex]\((a-b)(a^2+ab+b^2) = a^3 - b^3\)[/tex]
Identify [tex]\(a = x\)[/tex] and [tex]\(b = 1\)[/tex]:
[tex]\[ a^2 + ab + b^2 = x^2 + x + 1 \][/tex]
However, the given polynomial is [tex]\(x^2 - x + 1\)[/tex], which does not match [tex]\(x^2 + x + 1\)[/tex].
This product also does not fit the formula for difference of cubes.
3. [tex]\((x-1)\left(x^2+x+1\right)\)[/tex]
- Difference of Cubes Formula: [tex]\((a-b)(a^2+ab+b^2) = a^3 - b^3\)[/tex]
Identify [tex]\(a = x\)[/tex] and [tex]\(b = 1\)[/tex]:
[tex]\[ a^2 + ab + b^2 = x^2 + x + 1 \][/tex]
The given polynomial [tex]\(x^2+x+1\)[/tex] matches [tex]\(x^2 + ab + b^2\)[/tex].
This product fits the difference of cubes formula and results in [tex]\(x^3 - 1\)[/tex].
4. [tex]\((x+1)\left(x^2+x-1\right)\)[/tex]
- Sum of Cubes Formula: [tex]\((a+b)(a^2-ab+b^2) = a^3 + b^3\)[/tex]
Identify [tex]\(a = x\)[/tex] and [tex]\(b = 1\)[/tex]:
[tex]\[ a^2 - ab + b^2 = x^2 - x + 1 \][/tex]
However, the given polynomial is [tex]\(x^2 + x - 1\)[/tex], which does not match [tex]\(x^2 - x + 1\)[/tex].
This product does not fit either formula.
5. [tex]\((x+4)\left(x^2-4x+16\right)\)[/tex]
- Sum of Cubes Formula: [tex]\((a+b)(a^2-ab+b^2) = a^3 + b^3\)[/tex]
Identify [tex]\(a = x\)[/tex] and [tex]\(b = 4\)[/tex]:
[tex]\[ a^2 - ab + b^2 = x^2 - 4x + 16 \][/tex]
The given polynomial [tex]\(x^2-4x+16\)[/tex] matches [tex]\(a^2 - ab + b^2\)[/tex].
This product fits the sum of cubes formula and results in [tex]\(x^3 + 64\)[/tex].
6. [tex]\((x+4)\left(x^2+4x+16\right)\)[/tex]
- Sum of Cubes Formula: [tex]\((a+b)(a^2-ab+b^2) = a^3 + b^3\)[/tex]
Identify [tex]\(a = x\)[/tex] and [tex]\(b = 4\)[/tex]:
[tex]\[ a^2 - ab + b^2 = x^2 - 4x + 16 \][/tex]
However, the given polynomial is [tex]\(x^2 + 4x + 16\)[/tex], which does not match [tex]\(x^2 - 4x + 16\)[/tex].
This product does not fit either formula.
In conclusion, the products that result in a sum or difference of cubes are:
[tex]\[ (x-1)\left(x^2+x+1\right) \quad \text{and} \quad (x+4)\left(x^2-4x+16\right) \][/tex]
Thus, matching the results gives:
[tex]\[ [0, 0, 2, 0, 4, 0] \][/tex]
