To find the value of [tex]\( k \)[/tex] given that [tex]\( x + 2 \)[/tex] is a factor of the polynomial [tex]\( x^3 - k x^2 + 3 x + 7k \)[/tex], we can use the fact that if [tex]\( x + 2 \)[/tex] is a factor, then the polynomial must equal zero when [tex]\( x = -2 \)[/tex].
Here is the step-by-step solution:
1. Set up the polynomial: Start with the polynomial:
[tex]\[
P(x) = x^3 - k x^2 + 3 x + 7 k
\][/tex]
2. Substitute [tex]\( x = -2 \)[/tex] into the polynomial: Since [tex]\( x + 2 \)[/tex] is a factor, [tex]\( P(-2) \)[/tex] must equal zero.
[tex]\[
P(-2) = (-2)^3 - k(-2)^2 + 3(-2) + 7k
\][/tex]
3. Simplify the expression: Simplify each term separately.
[tex]\[
(-2)^3 = -8
\][/tex]
[tex]\[
-k(-2)^2 = -k(4) = -4k
\][/tex]
[tex]\[
3(-2) = -6
\][/tex]
[tex]\[
7k
\][/tex]
Put these simplified terms back into the equation:
[tex]\[
-8 - 4k - 6 + 7k = 0
\][/tex]
4. Combine like terms: Combine all the constants and the terms involving [tex]\( k \)[/tex].
[tex]\[
-8 - 6 = -14
\][/tex]
[tex]\[
-4k + 7k = 3k
\][/tex]
So the equation becomes:
[tex]\[
-14 + 3k = 0
\][/tex]
5. Solve for [tex]\( k \)[/tex]: Isolate [tex]\( k \)[/tex] on one side of the equation to find its value.
[tex]\[
3k = 14
\][/tex]
[tex]\[
k = \frac{14}{3}
\][/tex]
Therefore, the value of [tex]\( k \)[/tex] is:
[tex]\[
k = \frac{14}{3}
\][/tex]
