To determine which of the given options is a factor in the expression
[tex]\[
6z^4 - 4 + 9(y^3 + 3)
\][/tex]
it is helpful first to recognize the algebraic structure of the expression.
Let's break down and analyze each term:
The expression is:
[tex]\[
6z^4 - 4 + 9(y^3 + 3)
\][/tex]
Firstly, let's re-write the expression to make it more clear:
[tex]\[
6z^4 - 4 + 9y^3 + 27
\][/tex]
Now, let’s check each of the given options to determine if they are factors within this expression:
Option A: [tex]\(-4 + 9(y^3 + 3)\)[/tex]
[tex]\[
-4 + 9(y^3 + 3)
\][/tex]
Simplifies to:
[tex]\[
-4 + 9y^3 + 27
\][/tex]
Which results in:
[tex]\[
9y^3 + 23
\][/tex]
Clearly, this does not match any part of the original expression.
Option B: [tex]\(6z^4 - 4\)[/tex]
This term, [tex]\(6z^4 - 4\)[/tex], appears entirely independent of the term [tex]\(9(y^3 + 3)\)[/tex].
However, just because it is present in the original expression does not imply [tex]\(6z^4 - 4\)[/tex] is a factor. In the original expression, they are separate addends and thus not nested factors.
Option C: [tex]\(9(y^3 + 3)\)[/tex]
For this, factor out [tex]\(9(y^3 + 3)\)[/tex]:
[tex]\[
9(y^3 + 3) = 9y^3 + 27
\][/tex]
This matches the part of the expression [tex]\(9y^3 + 27\)[/tex].
Option D: [tex]\((y^3 + 3)\)[/tex]
Here we can factor out [tex]\(y^3 + 3\)[/tex] from [tex]\(9(y^3 + 3)\)[/tex]:
[tex]\[
9(y^3 + 3) = 9y^3 + 27
\][/tex]
As seen, [tex]\(y^3 + 3\)[/tex] indeed is nested as part of [tex]\(9(y^3 + 3)\)[/tex].
After considering each option, we observe:
- Options C and D correctly identify parts of the expression components as factors.
- Bearing in mind, the simplest form of the factor present within another nested factor fits our questions’ criteria.
Therefore, the best answer among the given choices that directly factors into the original expression is:
[tex]\[
D. \left(y^3 + 3\right)
\][/tex]
