The value of k, if (x-1) is factor of `2x^(3) + x^(2)-4x + k,` is -

Options

A. `-1`
B. `3`
C. `1`
D. 2

Answer

To find the value of \( k \) such that \( (x - 1) \) is a factor of the polynomial \( 2x^3 + x^2 - 4x + k \), we can use the Factor Theorem. According to the Factor Theorem, if \( (x - a) \) is a factor of a polynomial \( P(x) \), then \( P(a) = 0 \).

### Step-by-Step Solution:

1. **Identify the Polynomial**:
We have the polynomial \( P(x) = 2x^3 + x^2 - 4x + k \).

2. **Set Up the Equation**:
Since \( (x - 1) \) is a factor, we need to evaluate \( P(1) \) and set it equal to 0:
\[ P(1) = 2(1)^3 + (1)^2 - 4(1) + k = 0 \]

3. **Calculate \( P(1) \)**:
Substitute \( x = 1 \) into the polynomial:
\[ P(1) = 2(1) + 1 - 4 + k = 0 \]
Simplifying this gives:
\[ 2 + 1 - 4 + k = 0 \]
\[ -1 + k = 0 \]

4. **Solve for \( k \)**:
Rearranging the equation gives:
\[ k = 1 \]

5. **Conclusion**:
The value of \( k \) is \( 1 \).

### Final Answer:
\[ k = 1 \]