Question

which are sums of perfect cubes? check all that apply.

\\(8x^6 + 27\\)
\\(x^9 + 1\\)
\\(81x^3 + 16x^6\\)
\\(x^6 + x^3\\)
\\(27x^9 + x^{12}\\)
\\(9x^3 + 27x^9\\)

Explanation:

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<reasoning>

Define the conditions for a sum of perfect cubes

An algebraic expression is a sum of perfect cubes if it can be written in the form \(A^3 + B^3\), where both \(A\) and \(B\) are monomials with integer coefficients and integer exponents. For a monomial \(c x^n\) to be a perfect cube, its coefficient \(c\) must be a perfect integer cube (e.g., \(1, 8, 27, 64, \dots\)) and the exponent \(n\) of the variable must be a multiple of 3.

Analyze each option

Using the Perfect Cubes and Sum of Cubes knowledge points:

• Option 1: \(8x^6 + 27\)
• \(8x^6 = (2x^2)^3\) (since \(2^3 = 8\) and \((x^2)^3 = x^6\))
• \(27 = 3^3\)
• Both terms are perfect cubes, so this is a sum of perfect cubes.

• Option 2: \(x^9 + 1\)
• \(x^9 = (x^3)^3\)
• \(1 = 1^3\)
• Both terms are perfect cubes, so this is a sum of perfect cubes.

• Option 3: \(81x^3 + 16x^6\)
• \(81\) is not a perfect cube (\(4^3 = 64\), \(5^3 = 125\)).
• \(16\) is not a perfect cube.
• This is not a sum of perfect cubes.

• Option 4: \(x^6 + x^3\)
• \(x^6 = (x^2)^3\)
• \(x^3 = (x)^3\)
• Both terms are perfect cubes, so this is a sum of perfect cubes.

• Option 5: \(27x^9 + x^{12}\)
• \(27x^9 = (3x^3)^3\)
• \(x^{12} = (x^4)^3\)
• Both terms are perfect cubes, so this is a sum of perfect cubes.

• Option 6: \(9x^3 + 27x^9\)
• \(9\) is not a perfect cube.
• This is not a sum of perfect cubes.

</reasoning>

<answer>
<mcq-correct>(A) \(8x^6 + 27\)</mcq-correct>
<mcq-correct>(B) \(x^9 + 1\)</mcq-correct>
<mcq-option>(C) \(81x^3 + 16x^6\)</mcq-option>
<mcq-correct>(D) \(x^6 + x^3\)</mcq-correct>
<mcq-correct>(E) \(27x^9 + x^{12}\)</mcq-correct>
<mcq-option>(F) \(9x^3 + 27x^9\)</mcq-option>
</answer>

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Answer:

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<reasoning>

Define the conditions for a sum of perfect cubes

An algebraic expression is a sum of perfect cubes if it can be written in the form \(A^3 + B^3\), where both \(A\) and \(B\) are monomials with integer coefficients and integer exponents. For a monomial \(c x^n\) to be a perfect cube, its coefficient \(c\) must be a perfect integer cube (e.g., \(1, 8, 27, 64, \dots\)) and the exponent \(n\) of the variable must be a multiple of 3.

Analyze each option

Using the Perfect Cubes and Sum of Cubes knowledge points:

• Option 1: \(8x^6 + 27\)
• \(8x^6 = (2x^2)^3\) (since \(2^3 = 8\) and \((x^2)^3 = x^6\))
• \(27 = 3^3\)
• Both terms are perfect cubes, so this is a sum of perfect cubes.

• Option 2: \(x^9 + 1\)
• \(x^9 = (x^3)^3\)
• \(1 = 1^3\)
• Both terms are perfect cubes, so this is a sum of perfect cubes.

• Option 3: \(81x^3 + 16x^6\)
• \(81\) is not a perfect cube (\(4^3 = 64\), \(5^3 = 125\)).
• \(16\) is not a perfect cube.
• This is not a sum of perfect cubes.

• Option 4: \(x^6 + x^3\)
• \(x^6 = (x^2)^3\)
• \(x^3 = (x)^3\)
• Both terms are perfect cubes, so this is a sum of perfect cubes.

• Option 5: \(27x^9 + x^{12}\)
• \(27x^9 = (3x^3)^3\)
• \(x^{12} = (x^4)^3\)
• Both terms are perfect cubes, so this is a sum of perfect cubes.

• Option 6: \(9x^3 + 27x^9\)
• \(9\) is not a perfect cube.
• This is not a sum of perfect cubes.

</reasoning>

<answer>
<mcq-correct>(A) \(8x^6 + 27\)</mcq-correct>
<mcq-correct>(B) \(x^9 + 1\)</mcq-correct>
<mcq-option>(C) \(81x^3 + 16x^6\)</mcq-option>
<mcq-correct>(D) \(x^6 + x^3\)</mcq-correct>
<mcq-correct>(E) \(27x^9 + x^{12}\)</mcq-correct>
<mcq-option>(F) \(9x^3 + 27x^9\)</mcq-option>
</answer>

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