Certainly! Let's solve the problem step-by-step:
1. Understand the Problem: We are given a polynomial [tex]\( f(x) = x^3 - 4k^2 x^3 + 2x + 2k + 3 \)[/tex] and need to determine the value of [tex]\( k \)[/tex] given that [tex]\( x + 2k \)[/tex] is a factor of [tex]\( f(x) \)[/tex].
2. Factor Theorem: According to the Factor Theorem, if [tex]\( x + 2k \)[/tex] is a factor of [tex]\( f(x) \)[/tex], then [tex]\( f(-2k) = 0 \)[/tex].
3. Substitute [tex]\( x = -2k \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[
f(-2k) = (-2k)^3 - 4k^2 (-2k)^3 + 2(-2k) + 2k + 3
\][/tex]
4. Simplify the Expression:
[tex]\[
f(-2k) = -8k^3 - 4k^2 (-8k^3) - 4k + 2k + 3
\][/tex]
Simplifying further inside the parentheses:
[tex]\[
f(-2k) = -8k^3 - 4k^2 (-8k^3) - 2k + 3
\][/tex]
Further simplify the second term:
[tex]\[
-4k^2 (-8k^3) = 32k^5
\][/tex]
Putting everything back together, we get:
[tex]\[
f(-2k) = -8k^3 + 32k^5 - 2k + 3
\][/tex]
5. Set the Expression Equal to Zero:
[tex]\[
-8k^3 + 32k^5 - 2k + 3 = 0
\][/tex]
This is a polynomial equation in [tex]\( k \)[/tex]:
[tex]\[
32k^5 - 8k^3 - 2k + 3 = 0
\][/tex]
6. Solve the Polynomial Equation: The values of [tex]\( k \)[/tex] are the roots of the equation:
[tex]\[
32k^5 - 8k^3 - 2k + 3 = 0
\][/tex]
These roots can be expressed as:
[tex]\[
k = \text{CRootOf(32k^5 - 8k^3 - 2k + 3, n)}
\][/tex]
where [tex]\( n \)[/tex] ranges from 0 to 4, each representing a distinct root of the polynomial equation.
7. The Solutions: Therefore, the values of [tex]\( k \)[/tex] which satisfy the condition that [tex]\( x + 2k \)[/tex] is a factor of [tex]\( f(x) \)[/tex] are given by:
[tex]\[
k = \text{CRootOf(32k^5 - 8k^3 - 2k + 3, 0)}
\][/tex]
[tex]\[
k = \text{CRootOf(32k^5 - 8k^3 - 2k + 3, 1)}
\][/tex]
[tex]\[
k = \text{CRootOf(32k^5 - 8k^3 - 2k + 3, 2)}
\][/tex]
[tex]\[
k = \text{CRootOf(32k^5 - 8k^3 - 2k + 3, 3)}
\][/tex]
[tex]\[
k = \text{CRootOf(32k^5 - 8k^3 - 2k + 3, 4)}
\][/tex]
This completes our solution to the problem.
