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Answer :
To determine which expressions are sums of perfect cubes, let's break down each expression and see if it fits the pattern for a sum of cubes.
A sum of cubes follows the pattern:
[tex]\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \][/tex]
Let's examine each expression:
1. Expression: [tex]\( 8x^6 + 27 \)[/tex]
- [tex]\( 8x^6 \)[/tex] is [tex]\((2x^2)^3\)[/tex] and [tex]\( 27 \)[/tex] is [tex]\(3^3\)[/tex].
- This expression can be rewritten as [tex]\((2x^2)^3 + 3^3\)[/tex].
- Therefore, [tex]\( 8x^6 + 27 \)[/tex] is indeed a sum of cubes.
2. Expression: [tex]\( x^9 + 1 \)[/tex]
- [tex]\( x^9 \)[/tex] is [tex]\((x^3)^3\)[/tex] and [tex]\( 1 \)[/tex] is [tex]\(1^3\)[/tex].
- This can be written as [tex]\((x^3)^3 + 1^3\)[/tex].
- So, [tex]\( x^9 + 1 \)[/tex] is a sum of cubes.
3. Expression: [tex]\( 81x^3 + 16x^6 \)[/tex]
- [tex]\( 81x^3 \)[/tex] is [tex]\((3x)^3\)[/tex] and [tex]\( 16x^6 \)[/tex] is [tex]\((2x^2)^3\)[/tex].
- This can be written as [tex]\((2x^2)^3 + (3x)^3\)[/tex].
- So, [tex]\( 81x^3 + 16x^6 \)[/tex] is a sum of cubes.
4. Expression: [tex]\( x^6 + x^3 \)[/tex]
- This expression can be factored as [tex]\( x^3(x^3 + 1) \)[/tex], but it is not in the form of a sum of cubes.
- [tex]\( x^6 \)[/tex] and [tex]\( x^3 \)[/tex] cannot be expressed as single cubes that sum up.
- Therefore, [tex]\( x^6 + x^3 \)[/tex] is not a sum of cubes.
5. Expression: [tex]\( 27x^9 + x^{12} \)[/tex]
- [tex]\( 27x^9 \)[/tex] is [tex]\((3x^3)^3\)[/tex] and [tex]\( x^{12} \)[/tex] is [tex]\((x^4)^3\)[/tex].
- This can be rewritten as [tex]\((3x^3)^3 + (x^4)^3\)[/tex].
- So, [tex]\( 27x^9 + x^{12} \)[/tex] is a sum of cubes.
6. Expression: [tex]\( 9x^3 + 27x^9 \)[/tex]
- This expression can be factored as [tex]\( 9x^3(1 + 3x^6) \)[/tex], but it is not a straightforward sum of cubes.
- Neither of the terms fits the cube pattern [tex]\( a^3 + b^3 \)[/tex].
- Therefore, [tex]\( 9x^3 + 27x^9 \)[/tex] is not a sum of cubes.
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 are the expressions that fit the pattern of sums of perfect cubes.
A sum of cubes follows the pattern:
[tex]\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \][/tex]
Let's examine each expression:
1. Expression: [tex]\( 8x^6 + 27 \)[/tex]
- [tex]\( 8x^6 \)[/tex] is [tex]\((2x^2)^3\)[/tex] and [tex]\( 27 \)[/tex] is [tex]\(3^3\)[/tex].
- This expression can be rewritten as [tex]\((2x^2)^3 + 3^3\)[/tex].
- Therefore, [tex]\( 8x^6 + 27 \)[/tex] is indeed a sum of cubes.
2. Expression: [tex]\( x^9 + 1 \)[/tex]
- [tex]\( x^9 \)[/tex] is [tex]\((x^3)^3\)[/tex] and [tex]\( 1 \)[/tex] is [tex]\(1^3\)[/tex].
- This can be written as [tex]\((x^3)^3 + 1^3\)[/tex].
- So, [tex]\( x^9 + 1 \)[/tex] is a sum of cubes.
3. Expression: [tex]\( 81x^3 + 16x^6 \)[/tex]
- [tex]\( 81x^3 \)[/tex] is [tex]\((3x)^3\)[/tex] and [tex]\( 16x^6 \)[/tex] is [tex]\((2x^2)^3\)[/tex].
- This can be written as [tex]\((2x^2)^3 + (3x)^3\)[/tex].
- So, [tex]\( 81x^3 + 16x^6 \)[/tex] is a sum of cubes.
4. Expression: [tex]\( x^6 + x^3 \)[/tex]
- This expression can be factored as [tex]\( x^3(x^3 + 1) \)[/tex], but it is not in the form of a sum of cubes.
- [tex]\( x^6 \)[/tex] and [tex]\( x^3 \)[/tex] cannot be expressed as single cubes that sum up.
- Therefore, [tex]\( x^6 + x^3 \)[/tex] is not a sum of cubes.
5. Expression: [tex]\( 27x^9 + x^{12} \)[/tex]
- [tex]\( 27x^9 \)[/tex] is [tex]\((3x^3)^3\)[/tex] and [tex]\( x^{12} \)[/tex] is [tex]\((x^4)^3\)[/tex].
- This can be rewritten as [tex]\((3x^3)^3 + (x^4)^3\)[/tex].
- So, [tex]\( 27x^9 + x^{12} \)[/tex] is a sum of cubes.
6. Expression: [tex]\( 9x^3 + 27x^9 \)[/tex]
- This expression can be factored as [tex]\( 9x^3(1 + 3x^6) \)[/tex], but it is not a straightforward sum of cubes.
- Neither of the terms fits the cube pattern [tex]\( a^3 + b^3 \)[/tex].
- Therefore, [tex]\( 9x^3 + 27x^9 \)[/tex] is not a sum of cubes.
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 are the expressions that fit the pattern of sums of perfect cubes.
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