To determine the value of [tex]\( k \)[/tex] such that [tex]\((x - 2)\)[/tex] is a factor of the polynomial [tex]\( f(x) = 3x^3 - 8x^2 + 5x - k \)[/tex], we can use the Factor Theorem. The Factor Theorem states that if [tex]\( (x - a) \)[/tex] is a factor of a polynomial [tex]\( f(x) \)[/tex], then [tex]\( f(a) = 0 \)[/tex].
Given that [tex]\((x-2)\)[/tex] is a factor, it means that substituting [tex]\( x = 2 \)[/tex] into the polynomial should result in the polynomial being equal to 0. Therefore, we need to find [tex]\( k \)[/tex] such that:
[tex]\[
f(2) = 0
\][/tex]
Substitute [tex]\( x = 2 \)[/tex] into the polynomial [tex]\( f(x) \)[/tex]:
[tex]\[
f(2) = 3(2)^3 - 8(2)^2 + 5(2) - k
\][/tex]
Calculate each term separately:
[tex]\[
2^3 = 8 \implies 3(8) = 24
\][/tex]
[tex]\[
2^2 = 4 \implies 8(4) = 32
\][/tex]
[tex]\[
5(2) = 10
\][/tex]
Now, substitute these values back into the equation:
[tex]\[
f(2) = 24 - 32 + 10 - k
\][/tex]
Combine the constants:
[tex]\[
24 - 32 + 10 = 2
\][/tex]
Therefore, we have:
[tex]\[
f(2) = 2 - k
\][/tex]
Since [tex]\((x-2)\)[/tex] is a factor, [tex]\( f(2) = 0 \)[/tex]. Hence, we set up the equation:
[tex]\[
2 - k = 0
\][/tex]
Solve for [tex]\( k \)[/tex]:
[tex]\[
k = 2
\][/tex]
Thus, the value of [tex]\( k \)[/tex] that makes [tex]\((x-2)\)[/tex] a factor of [tex]\( f(x) = 3x^3 - 8x^2 + 5x - k \)[/tex] is:
[tex]\[
\boxed{2}
\][/tex]
