For a polynomial `p(x)` of degree `ge1, p(a)=0` , where a is a real number, then `(x-a)` is a factor of the polynomial `p(x)`
Find the value of k if `x-1` is a factor of `4x^(3)+3x^(2)-4x+k` .

Options

A. `0`
B. `1`
C. `-3`
D. `2`

Answer

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

### Step-by-Step Solution:

1. **Identify the polynomial and the factor**:
We have the polynomial \( p(x) = 4x^3 + 3x^2 - 4x + k \) and we know that \( x - 1 \) is a factor. This means we need to evaluate \( p(1) \).

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

3. **Calculate \( p(1) \)**:
\[ p(1) = 4(1) + 3(1) - 4 + k = 4 + 3 - 4 + k \]
Simplifying this gives:
\[ p(1) = 4 + 3 - 4 + k = 3 + k \]

4. **Set \( p(1) \) equal to 0**:
Since \( x - 1 \) is a factor, we set \( p(1) = 0 \):
\[ 3 + k = 0 \]

5. **Solve for \( k \)**:
\[ k = -3 \]

### Conclusion:
The value of \( k \) is \( -3 \).