Question

To factor the expression
8
𝑥
3
−
27
8x
3
−27, recognize it as a difference of cubes. The general form for factoring a difference of cubes is:

Answer 1

The general form for factoring a difference of cubes \(a^3 - b^3\) is:

\[
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
\]

For the expression \(8x^3 - 27\), we can identify \(8x^3\) and \(27\) as cubes:

- \(8x^3 = (2x)^3\)
- \(27 = 3^3\)

Thus, we can set \(a = 2x\) and \(b = 3\).

Now we can substitute \(a\) and \(b\) into the difference of cubes formula:

\[
8x^3 - 27 = (2x - 3)((2x)^2 + (2x)(3) + (3)^2)
\]

Now, simplify \(a^2 + ab + b^2\):

1. \((2x)^2 = 4x^2\)
2. \((2x)(3) = 6x\)
3. \((3)^2 = 9\)

Putting it all together, we have:

\[
8x^3 - 27 = (2x - 3)(4x^2 + 6x + 9)
\]

Thus, the factored form of the expression \(8x^3 - 27\) is:

\[
(2x - 3)(4x^2 + 6x + 9)
\]