To find a factor of the expression \(27x^3 + 512y^3\), we recognize that this expression can be factored using the sum of cubes formula. The sum of cubes formula states that:
\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]
We can write \(27x^3\) as \((3x)^3\) and \(512y^3\) as \((8y)^3\). Setting \(a = 3x\) and \(b = 8y\), we can apply the sum of cubes factorization:
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Calculate \(a + b\): \[ 3x + 8y \]
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Calculate \(a^2\): \[ (3x)^2 = 9x^2 \]
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Calculate \(-ab\): \[ -(3x)(8y) = -24xy \]
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Calculate \(b^2\): \[ (8y)^2 = 64y^2 \]
Putting it all together, we get:
\[ 27x^3 + 512y^3 = (3x + 8y)(9x^2 - 24xy + 64y^2) \]
Next, we look at the options given:
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3: This is not a factor of the expression we factored.
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3x − 8y: This is not the correct expression; we found \(3x + 8y\).
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9x^2 − 24xy + 64y^2: This corresponds to the second factor derived from our factoring process.
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9x^2 + 24xy + 64y^2: This is not one of the factors.
Thus, the correct answer is:
9x^2 − 24xy + 64y^2
1 answer