Answer :
To determine which expressions are sums of perfect cubes, we need to analyze each expression and see if it can be rewritten in the form of [tex]\(a^3 + b^3\)[/tex]. Recall that a sum of perfect cubes can be factored as [tex]\(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\)[/tex].
Let's examine each given expression:
1. [tex]\(8x^6 + 27\)[/tex]
- Here, [tex]\(8x^6 = (2x^2)^3\)[/tex] and [tex]\(27 = 3^3\)[/tex].
- This expression is a sum of perfect cubes: [tex]\((2x^2)^3 + 3^3\)[/tex].
2. [tex]\(x^9 + 1\)[/tex]
- [tex]\(x^9 = (x^3)^3\)[/tex] and [tex]\(1 = 1^3\)[/tex].
- This expression is a sum of perfect cubes: [tex]\((x^3)^3 + 1^3\)[/tex].
3. [tex]\(81x^3 + 16x^6\)[/tex]
- Rewriting: [tex]\(81x^3 = (3x)^3\)[/tex] and [tex]\(16x^6 = (2x^2)^3\)[/tex].
- This expression is a sum of perfect cubes: [tex]\((3x)^3 + (2x^2)^3\)[/tex].
4. [tex]\(x^6 + x^3\)[/tex]
- Factor as: [tex]\(x^3(x^3 + 1)\)[/tex].
- [tex]\(x^3 + 1\)[/tex] is a sum of cubes if we consider: [tex]\((x^3) + 1^3\)[/tex], but not the entire original expression directly as a sum of cubes.
5. [tex]\(27x^9 + x^{12}\)[/tex]
- [tex]\(27x^9 = (3x^3)^3\)[/tex] and [tex]\(x^{12} = (x^4)^3\)[/tex].
- This expression is a sum of perfect cubes: [tex]\((3x^3)^3 + (x^4)^3\)[/tex].
6. [tex]\(9x^3 + 27x^9\)[/tex]
- Factor as: [tex]\(9x^3(1 + 3x^6)\)[/tex].
- This expression cannot be directly rewritten as a sum of perfect cubes.
The expressions that are sums of perfect cubes are:
- [tex]\(8x^6 + 27\)[/tex]
- [tex]\(x^9 + 1\)[/tex]
- [tex]\(81x^3 + 16x^6\)[/tex]
- [tex]\(27x^9 + x^{12}\)[/tex]
These can all be rewritten in the form [tex]\(a^3 + b^3\)[/tex].
Let's examine each given expression:
1. [tex]\(8x^6 + 27\)[/tex]
- Here, [tex]\(8x^6 = (2x^2)^3\)[/tex] and [tex]\(27 = 3^3\)[/tex].
- This expression is a sum of perfect cubes: [tex]\((2x^2)^3 + 3^3\)[/tex].
2. [tex]\(x^9 + 1\)[/tex]
- [tex]\(x^9 = (x^3)^3\)[/tex] and [tex]\(1 = 1^3\)[/tex].
- This expression is a sum of perfect cubes: [tex]\((x^3)^3 + 1^3\)[/tex].
3. [tex]\(81x^3 + 16x^6\)[/tex]
- Rewriting: [tex]\(81x^3 = (3x)^3\)[/tex] and [tex]\(16x^6 = (2x^2)^3\)[/tex].
- This expression is a sum of perfect cubes: [tex]\((3x)^3 + (2x^2)^3\)[/tex].
4. [tex]\(x^6 + x^3\)[/tex]
- Factor as: [tex]\(x^3(x^3 + 1)\)[/tex].
- [tex]\(x^3 + 1\)[/tex] is a sum of cubes if we consider: [tex]\((x^3) + 1^3\)[/tex], but not the entire original expression directly as a sum of cubes.
5. [tex]\(27x^9 + x^{12}\)[/tex]
- [tex]\(27x^9 = (3x^3)^3\)[/tex] and [tex]\(x^{12} = (x^4)^3\)[/tex].
- This expression is a sum of perfect cubes: [tex]\((3x^3)^3 + (x^4)^3\)[/tex].
6. [tex]\(9x^3 + 27x^9\)[/tex]
- Factor as: [tex]\(9x^3(1 + 3x^6)\)[/tex].
- This expression cannot be directly rewritten as a sum of perfect cubes.
The expressions that are sums of perfect cubes are:
- [tex]\(8x^6 + 27\)[/tex]
- [tex]\(x^9 + 1\)[/tex]
- [tex]\(81x^3 + 16x^6\)[/tex]
- [tex]\(27x^9 + x^{12}\)[/tex]
These can all be rewritten in the form [tex]\(a^3 + b^3\)[/tex].
