Answer :
To determine which expressions are sums of perfect cubes, let's review what a sum of cubes looks like.
A sum of cubes typically follows the formula:
[tex]\[ a^3 + b^3 \][/tex]
When factored, it becomes:
[tex]\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \][/tex]
Let's evaluate each expression using these concepts:
1. [tex]\(8x^6 + 27\)[/tex]:
- Rewrite the terms as cubes:
[tex]\[ (2x^2)^3 + 3^3 \][/tex]
- This expression matches the form [tex]\( a^3 + b^3 \)[/tex] with [tex]\( a = 2x^2 \)[/tex] and [tex]\( b = 3 \)[/tex]. Thus, it is a sum of cubes.
2. [tex]\(x^9 + 1\)[/tex]:
- Rewrite the terms as cubes:
[tex]\[ (x^3)^3 + 1^3 \][/tex]
- This fits the form [tex]\( a^3 + b^3 \)[/tex] with [tex]\( a = x^3 \)[/tex] and [tex]\( b = 1 \)[/tex]. It is a sum of cubes.
3. [tex]\(81x^3 + 16x^6\)[/tex]:
- Factor out anything common:
[tex]\[ x^3(81 + 16x^3) \][/tex]
- Rewrite inside the parentheses:
[tex]\[ 81 + (2x)^3 = (3^4) + (2x)^3 \][/tex]
- This does not match the form of a sum of cubes directly; therefore, it's not a sum of cubes.
4. [tex]\(x^6 + x^3\)[/tex]:
- Factor out [tex]\( x^3 \)[/tex]:
[tex]\[ x^3(x^3 + 1) \][/tex]
- Rewrite inside the parentheses:
[tex]\[ x^3 + 1^3 \][/tex]
- This matches the form [tex]\( a^3 + b^3 \)[/tex] with [tex]\( a = x \)[/tex] and [tex]\( b = 1 \)[/tex], but the whole expression is not a perfect cube alone, although part of it is.
5. [tex]\(27x^9 + x^{12}\)[/tex]:
- Rewrite the terms as cubes:
[tex]\[ (3x^3)^3 + (x^4)^3 \][/tex]
- This expression matches the form [tex]\( a^3 + b^3 \)[/tex] with [tex]\( a = 3x^3 \)[/tex] and [tex]\( b = x^4 \)[/tex]. Thus, it is a sum of cubes.
6. [tex]\(9x^3 + 27x^9\)[/tex]:
- Factor out anything common:
[tex]\[ 9x^3(1 + 3x^6) \][/tex]
- This expression does not fit into the sum of cubes as the factor and remainder don't form perfect cubes directly.
Summary of which are sums of perfect cubes:
- [tex]\( 8x^6 + 27 \)[/tex]
- [tex]\( x^9 + 1 \)[/tex]
- [tex]\( 27x^9 + x^{12} \)[/tex]
These are the expressions that can be classified as sums of perfect cubes.
A sum of cubes typically follows the formula:
[tex]\[ a^3 + b^3 \][/tex]
When factored, it becomes:
[tex]\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \][/tex]
Let's evaluate each expression using these concepts:
1. [tex]\(8x^6 + 27\)[/tex]:
- Rewrite the terms as cubes:
[tex]\[ (2x^2)^3 + 3^3 \][/tex]
- This expression matches the form [tex]\( a^3 + b^3 \)[/tex] with [tex]\( a = 2x^2 \)[/tex] and [tex]\( b = 3 \)[/tex]. Thus, it is a sum of cubes.
2. [tex]\(x^9 + 1\)[/tex]:
- Rewrite the terms as cubes:
[tex]\[ (x^3)^3 + 1^3 \][/tex]
- This fits the form [tex]\( a^3 + b^3 \)[/tex] with [tex]\( a = x^3 \)[/tex] and [tex]\( b = 1 \)[/tex]. It is a sum of cubes.
3. [tex]\(81x^3 + 16x^6\)[/tex]:
- Factor out anything common:
[tex]\[ x^3(81 + 16x^3) \][/tex]
- Rewrite inside the parentheses:
[tex]\[ 81 + (2x)^3 = (3^4) + (2x)^3 \][/tex]
- This does not match the form of a sum of cubes directly; therefore, it's not a sum of cubes.
4. [tex]\(x^6 + x^3\)[/tex]:
- Factor out [tex]\( x^3 \)[/tex]:
[tex]\[ x^3(x^3 + 1) \][/tex]
- Rewrite inside the parentheses:
[tex]\[ x^3 + 1^3 \][/tex]
- This matches the form [tex]\( a^3 + b^3 \)[/tex] with [tex]\( a = x \)[/tex] and [tex]\( b = 1 \)[/tex], but the whole expression is not a perfect cube alone, although part of it is.
5. [tex]\(27x^9 + x^{12}\)[/tex]:
- Rewrite the terms as cubes:
[tex]\[ (3x^3)^3 + (x^4)^3 \][/tex]
- This expression matches the form [tex]\( a^3 + b^3 \)[/tex] with [tex]\( a = 3x^3 \)[/tex] and [tex]\( b = x^4 \)[/tex]. Thus, it is a sum of cubes.
6. [tex]\(9x^3 + 27x^9\)[/tex]:
- Factor out anything common:
[tex]\[ 9x^3(1 + 3x^6) \][/tex]
- This expression does not fit into the sum of cubes as the factor and remainder don't form perfect cubes directly.
Summary of which are sums of perfect cubes:
- [tex]\( 8x^6 + 27 \)[/tex]
- [tex]\( x^9 + 1 \)[/tex]
- [tex]\( 27x^9 + x^{12} \)[/tex]
These are the expressions that can be classified as sums of perfect cubes.
