The expression 2x^3-5x^2-16x+k has a remainder of -6 when divided by (x-4). Find the value of k. [4]

To find the value of \( k \) such that the polynomial \( 2x^3 - 5x^2 - 16x + k \) has a remainder of \(-6\) when divided by \( (x - 4) \), we can use the **Remainder Theorem**.

### Remainder Theorem:
The Remainder Theorem states that if a polynomial \( f(x) \) is divided by \( (x - a) \), the remainder is \( f(a) \).

### Steps to solve:
1. **Identify the polynomial**:  
   \( f(x) = 2x^3 - 5x^2 - 16x + k \)

2. **Use the Remainder Theorem**:  
   Since the polynomial is divided by \( (x - 4) \), the remainder when substituting \( x = 4 \) into the polynomial is given as \(-6\).

   So,  
   \[
   f(4) = 2(4)^3 - 5(4)^2 - 16(4) + k = -6
   \]

3. **Substitute \( x = 4 \) into the polynomial**:  
   Calculate each term:  
   \[
   2(4)^3 = 2 \times 64 = 128
   \]
   \[
   -5(4)^2 = -5 \times 16 = -80
   \]
   \[
   -16(4) = -64
   \]
   Now, substitute these values into the equation:
   \[
   128 - 80 - 64 + k = -6
   \]

4. **Simplify the equation**:
   \[
   128 - 80 - 64 = -16
   \]
   So,  
   \[
   -16 + k = -6
   \]

5. **Solve for \( k \)**:
   Add 16 to both sides:
   \[
   k = -6 + 16
   \]
   \[
   k = 10
   \]

### Final Answer:
The value of \( k \) is \( \boxed{10} \).

Would you like to see more details or have any questions?

Here are some related questions you might find interesting:
1. What is the Remainder Theorem and how does it apply to polynomial division?
2. How would you find the remainder of a polynomial divided by a quadratic divisor?
3. What is synthetic division, and how can it be used instead of the Remainder Theorem?
4. Can the Remainder Theorem be applied to any polynomial division?
5. What is the Factor Theorem, and how is it related to the Remainder Theorem?
6. How would the process change if the divisor was \( (x + 4) \) instead of \( (x - 4) \)?
7. What if the remainder were zero? What would that indicate about \( k \)?
8. How can you verify the result using direct polynomial division?

**Tip**: Always double-check your substitution and arithmetic when applying the Remainder Theorem to avoid simple errors.

Learn how to find the value of k in the polynomial 2x^3 - 5x^2 - 16x + k using polynomial division and the Remainder Theorem. This problem demonstrates advanced algebraic techniques and is suitable for high school students.

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