Answer :
To determine which of the given polynomials results in a sum or difference of cubes, we can use the fact that there are specific factorizations for the sum and difference of cubes:
- The sum of cubes: [tex]\( a^3 + b^3 = (a + b)(a^2 - ab + b^2) \)[/tex]
- The difference of cubes: [tex]\( a^3 - b^3 = (a - b)(a^2 + ab + b^2) \)[/tex]
Let's compare each given polynomial with these formulas.
### Polynomial 1: [tex]\( (x + 7)(x^2 - 7x + 14) \)[/tex]
To match this with the sum of cubes formula [tex]\( (a + b)(a^2 - ab + b^2) \)[/tex]:
- [tex]\( a = x \)[/tex]
- [tex]\( b = 7 \)[/tex]
Applying these values:
[tex]\[ (a + b)(a^2 - ab + b^2) = (x + 7)[x^2 - x \cdot 7 + 7^2] = (x + 7)(x^2 - 7x + 49) \][/tex]
Since [tex]\( (x^2 - 7x + 49) \neq (x^2 - 7x + 14) \)[/tex], this polynomial does not represent a sum of cubes.
### Polynomial 2: [tex]\( (x + 8)(x^2 + 8x + 64) \)[/tex]
To match this with the sum of cubes formula [tex]\( (a + b)(a^2 - ab + b^2) \)[/tex]:
- [tex]\( a = x \)[/tex]
- [tex]\( b = 8 \)[/tex]
Applying these values:
[tex]\[ (a + b)(a^2 - ab + b^2) = (x + 8)[x^2 - x \cdot 8 + 8^2] = (x + 8)(x^2 - 8x + 64) \][/tex]
Since [tex]\( (x^2 + 8x + 64) \neq (x^2 - 8x + 64) \)[/tex], this polynomial does not represent a sum of cubes either.
### Polynomial 3: [tex]\( (x - 9)(x^2 + 9x + 81) \)[/tex]
To match this with the difference of cubes formula [tex]\( (a - b)(a^2 + ab + b^2) \)[/tex]:
- [tex]\( a = x \)[/tex]
- [tex]\( b = 9 \)[/tex]
Applying these values:
[tex]\[ (a - b)(a^2 + ab + b^2) = (x - 9)[x^2 + x \cdot 9 + 9^2] = (x - 9)(x^2 + 9x + 81) \][/tex]
This polynomial matches the formula for the difference of cubes exactly.
### Polynomial 4: [tex]\( (x - 10)(x^2 - 10x + 100) \)[/tex]
To match this with the difference of cubes formula [tex]\( (a - b)(a^2 + ab + b^2) \)[/tex]:
- [tex]\( a = x \)[/tex]
- [tex]\( b = 10 \)[/tex]
Applying these values:
[tex]\[ (a - b)(a^2 + ab + b^2) = (x - 10)[x^2 + x \cdot 10 + 10^2] = (x - 10)(x^2 + 10x + 100) \][/tex]
Since [tex]\( (x^2 - 10x + 100) \neq (x^2 + 10x + 100) \)[/tex], this polynomial does not represent a difference of cubes.
Therefore, the product that results in a sum or difference of cubes is:
[tex]\[ (x - 9)(x^2 + 9x + 81) \][/tex]
- The sum of cubes: [tex]\( a^3 + b^3 = (a + b)(a^2 - ab + b^2) \)[/tex]
- The difference of cubes: [tex]\( a^3 - b^3 = (a - b)(a^2 + ab + b^2) \)[/tex]
Let's compare each given polynomial with these formulas.
### Polynomial 1: [tex]\( (x + 7)(x^2 - 7x + 14) \)[/tex]
To match this with the sum of cubes formula [tex]\( (a + b)(a^2 - ab + b^2) \)[/tex]:
- [tex]\( a = x \)[/tex]
- [tex]\( b = 7 \)[/tex]
Applying these values:
[tex]\[ (a + b)(a^2 - ab + b^2) = (x + 7)[x^2 - x \cdot 7 + 7^2] = (x + 7)(x^2 - 7x + 49) \][/tex]
Since [tex]\( (x^2 - 7x + 49) \neq (x^2 - 7x + 14) \)[/tex], this polynomial does not represent a sum of cubes.
### Polynomial 2: [tex]\( (x + 8)(x^2 + 8x + 64) \)[/tex]
To match this with the sum of cubes formula [tex]\( (a + b)(a^2 - ab + b^2) \)[/tex]:
- [tex]\( a = x \)[/tex]
- [tex]\( b = 8 \)[/tex]
Applying these values:
[tex]\[ (a + b)(a^2 - ab + b^2) = (x + 8)[x^2 - x \cdot 8 + 8^2] = (x + 8)(x^2 - 8x + 64) \][/tex]
Since [tex]\( (x^2 + 8x + 64) \neq (x^2 - 8x + 64) \)[/tex], this polynomial does not represent a sum of cubes either.
### Polynomial 3: [tex]\( (x - 9)(x^2 + 9x + 81) \)[/tex]
To match this with the difference of cubes formula [tex]\( (a - b)(a^2 + ab + b^2) \)[/tex]:
- [tex]\( a = x \)[/tex]
- [tex]\( b = 9 \)[/tex]
Applying these values:
[tex]\[ (a - b)(a^2 + ab + b^2) = (x - 9)[x^2 + x \cdot 9 + 9^2] = (x - 9)(x^2 + 9x + 81) \][/tex]
This polynomial matches the formula for the difference of cubes exactly.
### Polynomial 4: [tex]\( (x - 10)(x^2 - 10x + 100) \)[/tex]
To match this with the difference of cubes formula [tex]\( (a - b)(a^2 + ab + b^2) \)[/tex]:
- [tex]\( a = x \)[/tex]
- [tex]\( b = 10 \)[/tex]
Applying these values:
[tex]\[ (a - b)(a^2 + ab + b^2) = (x - 10)[x^2 + x \cdot 10 + 10^2] = (x - 10)(x^2 + 10x + 100) \][/tex]
Since [tex]\( (x^2 - 10x + 100) \neq (x^2 + 10x + 100) \)[/tex], this polynomial does not represent a difference of cubes.
Therefore, the product that results in a sum or difference of cubes is:
[tex]\[ (x - 9)(x^2 + 9x + 81) \][/tex]
