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 \( A \) is a solution, we substitute \( x \) with \( A \) in the equation:
\[ A^2 - A(A + B) + k = 0 \]

2. **Simplify the equation**: Now, we simplify the left-hand side:
\[ A^2 - A^2 - AB + k = 0 \]
Here, \( A(A + B) = A^2 + AB \), so when we distribute the negative sign, \( -A^2 - AB \) results in:
\[ -AB + k = 0 \]

3. **Rearrange the equation to solve for \( k \)**: We can isolate \( k \) by adding \( AB \) to both sides:
\[ k = AB \]

### Final Answer:
Thus, the value of \( k \) is:
\[ \boxed{AB} \]