If x=a is a solution of the quadratic polynomial `x^2-x(a+b)+k=0`, then find the value of k:

Options

A. a
B. b
C. ab
D. `a^2b^2`

Answer

To find the value of \( k \) in the quadratic polynomial \( x^2 - x(a + b) + k = 0 \) given that \( x = a \) is a solution, we can follow these steps:

### Step-by-Step Solution:

1. **Substitute \( x = a \) into the polynomial**:
Since \( x = a \) is a solution, we substitute \( a \) into the polynomial:
\[ f(a) = a^2 - a(a + b) + k \]

2. **Set the polynomial equal to zero**:
Since \( x = a \) is a root, we have:
\[ f(a) = 0 \]
Therefore, we can write:
\[ a^2 - a(a + b) + k = 0 \]

3. **Simplify the equation**:
Expand the term \( -a(a + b) \):
\[ -a(a + b) = -a^2 - ab \]
Now, substitute this back into the equation:
\[ a^2 - a^2 - ab + k = 0 \]

4. **Combine like terms**:
The \( a^2 \) terms cancel out:
\[ -ab + k = 0 \]

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

### Final Answer:
The value of \( k \) is:
\[ \boxed{ab} \]