To determine the value of [tex]\( k \)[/tex] such that [tex]\( x+2 \)[/tex] is a factor of the polynomial [tex]\( x^3 - 6x^2 - 11x + k \)[/tex], we will use the Factor Theorem. The Factor Theorem states that [tex]\( x+c \)[/tex] is a factor of a polynomial [tex]\( P(x) \)[/tex] if and only if [tex]\( P(-c) = 0 \)[/tex].
1. According to the Factor Theorem, if [tex]\( x+2 \)[/tex] is a factor of [tex]\( x^3 - 6x^2 - 11x + k \)[/tex], then we must have:
[tex]\[
f(-2) = 0
\][/tex]
Where [tex]\( f(x) = x^3 - 6x^2 - 11x + k \)[/tex].
2. Substitute [tex]\( x = -2 \)[/tex] into the polynomial [tex]\( f(x) \)[/tex]:
[tex]\[
f(-2) = (-2)^3 - 6(-2)^2 - 11(-2) + k
\][/tex]
3. Calculate each term individually:
[tex]\[
(-2)^3 = -8
\][/tex]
[tex]\[
- 6(-2)^2 = -6(4) = -24
\][/tex]
[tex]\[
- 11(-2) = 22
\][/tex]
4. Now, combine these values:
[tex]\[
f(-2) = -8 - 24 + 22 + k = 0
\][/tex]
5. Simplify the equation:
[tex]\[
-8 - 24 + 22 + k = 0
\][/tex]
[tex]\[
-8 - 24 = -32
\][/tex]
[tex]\[
-32 + 22 = -10
\][/tex]
Therefore, we have:
[tex]\[
-10 + k = 0
\][/tex]
6. Solving for [tex]\( k \)[/tex] gives:
[tex]\[
k = 10
\][/tex]
Hence, the value of [tex]\( k \)[/tex] is [tex]\( \boxed{10} \)[/tex].
