Question

Factor

8z3+27
8
𝑧
3
+
27
(1 point)
Responses

(2z+3)(4z2−12z+9)
(
2
𝑧
+
3
)
(
4
𝑧
2
−
12
𝑧
+
9
)
open paren 2 z plus 3 close paren times open paren 4 z squared minus 12 z plus 9 close paren

(2z−3)(4z2−6z−9)
(
2
𝑧
−
3
)
(
4
𝑧
2
−
6
𝑧
−
9
)
open paren 2 z minus 3 close paren times open paren 4 z squared minus 6 z minus 9 close paren

(2z−3)(4z2−12z−9)
(
2
𝑧
−
3
)
(
4
𝑧
2
−
12
𝑧
−
9
)
open paren 2 z minus 3 close paren times open paren 4 z squared minus 12 z minus 9 close paren

(2z+3)(4z2−6z+9)

Answer 1

To factor the expression \(8z^3 + 27\), we recognize that it is a sum of cubes. The formula for factoring a sum of cubes, \(a^3 + b^3\), is given by:

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

In this case:

- \(a = 2z\) (since \((2z)^3 = 8z^3\))
- \(b = 3\) (since \(3^3 = 27\))

Now we can apply the sum of cubes formula:

1. Identify \(a + b\):
\[
2z + 3
\]

2. Calculate \(a^2\):
\[
(2z)^2 = 4z^2
\]

3. Calculate \(ab\):
\[
2z \cdot 3 = 6z
\]

4. Calculate \(b^2\):
\[
3^2 = 9
\]

Next, we put it all together into the formula:

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

Thus, the correct factored form of \(8z^3 + 27\) is:

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

The answer choice that reflects this factored form is:

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

Therefore, the correct answer is:

\((2z + 3)(4z^2 - 6z + 9)\)