If `(x-2)` is a factor of polynomial `x^(2) + kx+4`. Find the value of k

Answer

To find the value of \( k \) such that \( (x - 2) \) is a factor of the polynomial \( x^2 + kx + 4 \), we can follow these steps:

### Step 1: Understand the Factor Condition
Since \( (x - 2) \) is a factor of the polynomial, it means that when we substitute \( x = 2 \) into the polynomial, the result should be zero.

### Step 2: Substitute \( x = 2 \) into the Polynomial
We substitute \( x = 2 \) into the polynomial \( f(x) = x^2 + kx + 4 \):
\[ f(2) = 2^2 + k(2) + 4 \]

### Step 3: Simplify the Expression
Now, simplify the expression:
\[ f(2) = 4 + 2k + 4 \]
\[ f(2) = 8 + 2k \]

### Step 4: Set the Expression to Zero
Since \( (x - 2) \) is a factor, we set \( f(2) = 0 \):
\[ 8 + 2k = 0 \]

### Step 5: Solve for \( k \)
Now, solve for \( k \):
\[ 2k = -8 \]
\[ k = \frac{-8}{2} \]
\[ k = -4 \]

### Conclusion
Thus, the value of \( k \) is \( -4 \).

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