If `(x+k)` is a common factor of `(x^(2)+px+q)` are `(x^(2)+lx+m)`, then the value of k is :

Options

A. `l+p`
B. `m-q`
C. `(l-p)/(m-q)`
D. `(m-q)/(l-p)`

Answer

To find the value of \( k \) given that \( (x + k) \) is a common factor of the polynomials \( (x^2 + px + q) \) and \( (x^2 + lx + m) \), we can follow these steps:

### Step 1: Set up the equations
Since \( (x + k) \) is a common factor, substituting \( x = -k \) into both polynomials should yield zero:
1. For the first polynomial:
\[ (-k)^2 + p(-k) + q = 0 \]
This simplifies to:
\[ k^2 - pk + q = 0 \quad (1) \]

2. For the second polynomial:
\[ (-k)^2 + l(-k) + m = 0 \]
This simplifies to:
\[ k^2 - lk + m = 0 \quad (2) \]

### Step 2: Equate the two equations
From equations (1) and (2), we can set them equal to each other since both equal zero:
\[ k^2 - pk + q = k^2 - lk + m \]

### Step 3: Simplify the equation
Cancelling \( k^2 \) from both sides gives:
\[ -pk + q = -lk + m \]
Rearranging this, we get:
\[ lk - pk = m - q \]

### Step 4: Factor out \( k \)
Factoring \( k \) out from the left side:
\[ k(l - p) = m - q \]

### Step 5: Solve for \( k \)
Now, we can solve for \( k \):
\[ k = \frac{m - q}{l - p} \]

### Conclusion
Thus, the value of \( k \) is:
\[ \boxed{\frac{m - q}{l - p}} \]