Question

Which of the following is a factor of 27x3 + 512y3?

3
3x − 8y
9x2 − 24xy + 64y2
9x2 + 24xy + 64y2

Answer 1

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:

1. Calculate \(a + b\):
\[
3x + 8y
\]

2. Calculate \(a^2\):
\[
(3x)^2 = 9x^2
\]

3. Calculate \(-ab\):
\[
-(3x)(8y) = -24xy
\]

4. 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:

- **3**: This is not a factor of the expression we factored.

- **3x − 8y**: This is not the correct expression; we found \(3x + 8y\).

- **9x^2 − 24xy + 64y^2**: This corresponds to the second factor derived from our factoring process.

- **9x^2 + 24xy + 64y^2**: This is not one of the factors.

Thus, the correct answer is:

**9x^2 − 24xy + 64y^2**