Answer :
Let's determine which of the given expressions are factors of the polynomial [tex]\( f(x) = 3x^3 - 10x^2 - 8x \)[/tex].
### Step 1: Check if [tex]\( x \)[/tex] is a factor
The expression [tex]\( x = 0 \)[/tex].
If [tex]\( x \)[/tex] is a factor, then [tex]\( f(0) \)[/tex] should be zero:
[tex]\[ f(0) = 3(0)^3 - 10(0)^2 - 8(0) = 0 \][/tex]
Since [tex]\( f(0) = 0 \)[/tex], [tex]\( x \)[/tex] is a factor of the polynomial.
### Step 2: Check if [tex]\( 3x - 2 \)[/tex] is a factor
The expression [tex]\( 3x - 2 \)[/tex]. To check if this is a factor, we substitute [tex]\( x = \frac{2}{3} \)[/tex]. If [tex]\( f\left(\frac{2}{3}\right) = 0 \)[/tex], then [tex]\( 3x - 2 \)[/tex] is a factor.
Calculate [tex]\( f\left(\frac{2}{3}\right) \)[/tex]:
[tex]\[ f\left(\frac{2}{3}\right) = 3\left(\frac{2}{3}\right)^3 - 10\left(\frac{2}{3}\right)^2 - 8\left(\frac{2}{3}\right) \][/tex]
[tex]\[ = 3 \left( \frac{8}{27} \right) - 10 \left( \frac{4}{9} \right) - 8 \left( \frac{2}{3} \right) \][/tex]
[tex]\[ = \frac{24}{27} - \frac{40}{9} - \frac{16}{3} \][/tex]
[tex]\[ = \frac{24}{27} - \frac{120}{27} - \frac{144}{27} \][/tex]
[tex]\[ = \frac{24 - 120 - 144}{27} \][/tex]
[tex]\[ = \frac{-240}{27} \][/tex]
[tex]\[ \neq 0 \][/tex]
Since [tex]\( f\left(\frac{2}{3}\right) \neq 0 \)[/tex], [tex]\( 3x - 2 \)[/tex] is not a factor of the polynomial.
### Step 3: Check if [tex]\( x - 4 \)[/tex] is a factor
The expression [tex]\( x - 4 \)[/tex]. To check if this is a factor, we substitute [tex]\( x = 4 \)[/tex]. If [tex]\( f(4) = 0 \)[/tex], then [tex]\( x - 4 \)[/tex] is a factor.
Calculate [tex]\( f(4) \)[/tex]:
[tex]\[ f(4) = 3(4)^3 - 10(4)^2 - 8(4) \][/tex]
[tex]\[ = 3(64) - 10(16) - 8(4) \][/tex]
[tex]\[ = 192 - 160 - 32 \][/tex]
[tex]\[ = 0 \][/tex]
Since [tex]\( f(4) = 0 \)[/tex], [tex]\( x - 4 \)[/tex] is a factor of the polynomial.
### Conclusion
Based on our calculations:
- [tex]\( x \)[/tex] is a factor.
- [tex]\( 3x - 2 \)[/tex] is not a factor.
- [tex]\( x - 4 \)[/tex] is a factor.
The correct answer is:
[tex]\[ \boxed{\text{F. I only}} \][/tex]
### Step 1: Check if [tex]\( x \)[/tex] is a factor
The expression [tex]\( x = 0 \)[/tex].
If [tex]\( x \)[/tex] is a factor, then [tex]\( f(0) \)[/tex] should be zero:
[tex]\[ f(0) = 3(0)^3 - 10(0)^2 - 8(0) = 0 \][/tex]
Since [tex]\( f(0) = 0 \)[/tex], [tex]\( x \)[/tex] is a factor of the polynomial.
### Step 2: Check if [tex]\( 3x - 2 \)[/tex] is a factor
The expression [tex]\( 3x - 2 \)[/tex]. To check if this is a factor, we substitute [tex]\( x = \frac{2}{3} \)[/tex]. If [tex]\( f\left(\frac{2}{3}\right) = 0 \)[/tex], then [tex]\( 3x - 2 \)[/tex] is a factor.
Calculate [tex]\( f\left(\frac{2}{3}\right) \)[/tex]:
[tex]\[ f\left(\frac{2}{3}\right) = 3\left(\frac{2}{3}\right)^3 - 10\left(\frac{2}{3}\right)^2 - 8\left(\frac{2}{3}\right) \][/tex]
[tex]\[ = 3 \left( \frac{8}{27} \right) - 10 \left( \frac{4}{9} \right) - 8 \left( \frac{2}{3} \right) \][/tex]
[tex]\[ = \frac{24}{27} - \frac{40}{9} - \frac{16}{3} \][/tex]
[tex]\[ = \frac{24}{27} - \frac{120}{27} - \frac{144}{27} \][/tex]
[tex]\[ = \frac{24 - 120 - 144}{27} \][/tex]
[tex]\[ = \frac{-240}{27} \][/tex]
[tex]\[ \neq 0 \][/tex]
Since [tex]\( f\left(\frac{2}{3}\right) \neq 0 \)[/tex], [tex]\( 3x - 2 \)[/tex] is not a factor of the polynomial.
### Step 3: Check if [tex]\( x - 4 \)[/tex] is a factor
The expression [tex]\( x - 4 \)[/tex]. To check if this is a factor, we substitute [tex]\( x = 4 \)[/tex]. If [tex]\( f(4) = 0 \)[/tex], then [tex]\( x - 4 \)[/tex] is a factor.
Calculate [tex]\( f(4) \)[/tex]:
[tex]\[ f(4) = 3(4)^3 - 10(4)^2 - 8(4) \][/tex]
[tex]\[ = 3(64) - 10(16) - 8(4) \][/tex]
[tex]\[ = 192 - 160 - 32 \][/tex]
[tex]\[ = 0 \][/tex]
Since [tex]\( f(4) = 0 \)[/tex], [tex]\( x - 4 \)[/tex] is a factor of the polynomial.
### Conclusion
Based on our calculations:
- [tex]\( x \)[/tex] is a factor.
- [tex]\( 3x - 2 \)[/tex] is not a factor.
- [tex]\( x - 4 \)[/tex] is a factor.
The correct answer is:
[tex]\[ \boxed{\text{F. I only}} \][/tex]
