When the polynomial `2x^(3)-kx^(2)+(5k-3)x-8` is divided by x-2 the remainder is 14. Find the value of 'k'.

Answer

To solve the problem, we need to find the value of \( k \) such that when the polynomial \( P(x) = 2x^3 - kx^2 + (5k - 3)x - 8 \) is divided by \( x - 2 \), the remainder is 14. We can use the Remainder Theorem, which states that the remainder of the division of a polynomial \( P(x) \) by \( x - a \) is \( P(a) \).

### Step-by-Step Solution:

1. **Identify the polynomial and the divisor**:
The polynomial is given as:
\[ P(x) = 2x^3 - kx^2 + (5k - 3)x - 8 \]
The divisor is \( x - 2 \).

2. **Apply the Remainder Theorem**:
According to the Remainder Theorem, we need to evaluate \( P(2) \) and set it equal to the given remainder, which is 14.
\[ P(2) = 2(2)^3 - k(2)^2 + (5k - 3)(2) - 8 \]

3. **Calculate \( P(2) \)**:
Substitute \( x = 2 \) into the polynomial:
\[ P(2) = 2(8) - k(4) + (5k - 3)(2) - 8 \]
Simplifying this:
\[ P(2) = 16 - 4k + (10k - 6) - 8 \]
Combine like terms:
\[ P(2) = 16 - 8 - 6 + 10k - 4k \]
\[ P(2) = 2 + 6k \]

4. **Set the expression equal to the remainder**:
We know that \( P(2) = 14 \):
\[ 2 + 6k = 14 \]

5. **Solve for \( k \)**:
Subtract 2 from both sides:
\[ 6k = 12 \]
Divide by 6:
\[ k = 2 \]

### Final Answer:
The value of \( k \) is \( 2 \).