9. Determine the value of k so that x 2 is a factor of x3-  2kx2 +6x - 4.

To determine the value of \( k \) such that \( x - 2 \) is a factor of the polynomial \( x^3 - 2kx^2 + 6x - 4 \), we need to use the Factor Theorem. According to the Factor Theorem, \( x - 2 \) is a factor of the polynomial if and only if the polynomial evaluates to zero when \( x = 2 \).

Given the polynomial:
\[ f(x) = x^3 - 2kx^2 + 6x - 4 \]

we need to find \( k \) such that \( f(2) = 0 \).

### Step-by-Step Solution:

1. **Substitute \( x = 2 \) into the polynomial:**
   \[
   f(2) = (2)^3 - 2k(2)^2 + 6(2) - 4
   \]

2. **Simplify the expression:**
   \[
   f(2) = 8 - 2k(4) + 12 - 4
   \]
   \[
   f(2) = 8 - 8k + 12 - 4
   \]
   \[
   f(2) = 16 - 8k
   \]

3. **Set the expression equal to zero and solve for \( k \):**
   \[
   16 - 8k = 0
   \]
   \[
   8k = 16
   \]
   \[
   k = 2
   \]

### Conclusion:

The value of \( k \) that makes \( x - 2 \) a factor of \( x^3 - 2kx^2 + 6x - 4 \) is \( \boxed{2} \).

Would you like any details or have any questions? Here are some possible follow-up questions:

1. How does the Factor Theorem work?
2. Can you show another example using the Factor Theorem?
3. What if the polynomial had more than one unknown coefficient?
4. How do you perform polynomial long division?
5. Could you demonstrate synthetic division for this problem?

**Tip:** When using the Factor Theorem, always verify your solution by substituting the found value back into the polynomial to ensure it evaluates to zero.

Learn how to determine the value of k such that x - 2 is a factor of the polynomial x^3 - 2kx^2 + 6x - 4. This problem involves applying the Factor Theorem and solving for an unknown coefficient. Suitable for students in Grades 10-12.

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