The value of k for which `(x-1)` is a factor of `9x^(2)+kx-18` is ____

Options

A. 9
B. 5
C. `-9`
D. 0

Answer

To find the value of \( k \) for which \( (x-1) \) is a factor of the polynomial \( 9x^2 + kx - 18 \), we can use the Factor Theorem. According to the Factor Theorem, if \( (x - 1) \) is a factor of the polynomial, then substituting \( x = 1 \) into the polynomial should yield a value of 0.

### Step-by-step Solution:

1. **Substitute \( x = 1 \) into the polynomial:**
\[ 9(1)^2 + k(1) - 18 = 0 \]

2. **Simplify the equation:**
\[ 9 \cdot 1 + k - 18 = 0 \]
\[ 9 + k - 18 = 0 \]

3. **Combine like terms:**
\[ k - 9 = 0 \]

4. **Solve for \( k \):**
\[ k = 9 \]

Thus, the value of \( k \) for which \( (x-1) \) is a factor of \( 9x^2 + kx - 18 \) is \( \boxed{9} \).