To determine which option is a factor in the given expression [tex]\(6z^4 - 4 + 9(y^3 + 3)\)[/tex], let's systematically check which options can divide the whole expression without leaving a remainder.
### Expression:
[tex]\[ 6z^4 - 4 + 9(y^3 + 3) \][/tex]
### Options:
1. [tex]\( 9(y^3 + 3) \)[/tex]
2. [tex]\( 6z^4 - 4 \)[/tex]
3. [tex]\( y^3 + 3 \)[/tex]
4. [tex]\( -4 + 9(y^3 + 3) \)[/tex]
#### Step-by-Step Analysis:
1. Option A: [tex]\(9(y^3 + 3)\)[/tex]
Substitute and check if dividing the whole expression by [tex]\(9(y^3 + 3)\)[/tex] results in an integer:
[tex]\[
\frac{6z^4 - 4 + 9(y^3 + 3)}{9(y^3 + 3)}
\][/tex]
Evaluating this expression, we do not get an integer. Hence, [tex]\(9(y^3 + 3)\)[/tex] is not a factor.
2. Option B: [tex]\(6z^4 - 4\)[/tex]
Substitute and check if dividing the whole expression by [tex]\(6z^4 - 4\)[/tex] results in an integer:
[tex]\[
\frac{6z^4 - 4 + 9(y^3 + 3)}{6z^4 - 4}
\][/tex]
Evaluating this, we see that the result is not an integer. Hence, [tex]\(6z^4 - 4\)[/tex] is not a factor.
3. Option C: [tex]\((y^3 + 3)\)[/tex]
Substitute and check if dividing the whole expression by [tex]\(y^3 + 3\)[/tex] results in an integer:
[tex]\[
\frac{6z^4 - 4 + 9(y^3 + 3)}{y^3 + 3}
\][/tex]
Evaluating this, we do not get an integer. Hence, [tex]\(y^3 + 3\)[/tex] is not a factor.
4. Option D: [tex]\(-4 + 9(y^3 + 3)\)[/tex]
Substitute and check if dividing the whole expression by [tex]\(-4 + 9(y^3 + 3)\)[/tex] results in an integer:
[tex]\[
\frac{6z^4 - 4 + 9(y^3 + 3)}{-4 + 9(y^3 + 3)}
\][/tex]
Evaluating this, we see that the result is not an integer. Hence, [tex]\(-4 + 9(y^3 + 3)\)[/tex] is not a factor.
Since none of the above options properly divide the expression without remainder, the expression [tex]\(6z^4 - 4 + 9(y^3 + 3)\)[/tex] does not have any of the given options as a factor.
Thus, the correct answer is None of the given options is a factor.
