To determine the value of [tex]\( k \)[/tex], given that [tex]\( (n-2) \)[/tex] is a factor of the polynomial equation [tex]\( 2n^3 + 3n^2 - kx + 4 = 0 \)[/tex], we can use the fact that if [tex]\( (n-2) \)[/tex] is a factor, then the polynomial must equal zero when [tex]\( n = 2 \)[/tex].
Let's substitute [tex]\( n = 2 \)[/tex] into the polynomial equation:
[tex]\[
2(2)^3 + 3(2)^2 - k(2) + 4 = 0
\][/tex]
Now, calculate each term step-by-step:
[tex]\[
2(2)^3 = 2 \cdot 8 = 16
\][/tex]
[tex]\[
3(2)^2 = 3 \cdot 4 = 12
\][/tex]
[tex]\[
-k(2) = -2k
\][/tex]
[tex]\[
4 = 4
\][/tex]
Now we substitute these values back into the polynomial equation:
[tex]\[
16 + 12 - 2k + 4 = 0
\][/tex]
Combine the constants:
[tex]\[
16 + 12 + 4 = 32
\][/tex]
So the equation simplifies to:
[tex]\[
32 - 2k = 0
\][/tex]
To solve for [tex]\( k \)[/tex], move [tex]\( 2k \)[/tex] to the left side:
[tex]\[
32 = 2k
\][/tex]
Divide both sides by 2:
[tex]\[
k = \frac{32}{2} = 16
\][/tex]
So the value of [tex]\( k \)[/tex] should be [tex]\( 16 \)[/tex], but none of the given options correspond to [tex]\( 16 \)[/tex]. Because the polynomial equation provided contained a term involving [tex]\( x \)[/tex], if it was a misread or misinterpretation, we could double check what polynomial was supposed to be calculated.
Hence, we check the options provided and re-confirm the scenario. If still there's a discrepancy within the task context itself then further context might be provided or clarified.
But following numeric calculation of provided options:
Considering the factorization expectation and solving appropriately the approach steps matches conceptual solution unless the term sign/expressions adds different [tex]\(x\)[/tex] specific including term incorrect interpreting specific as [tex]\(n\)[/tex].
For validation on closer stepwise ensures calculation for correctness approach and exact polynomial might adjusted known inconsistency in problem.
