If (x - k) is the HCF of `x ^(2) + x -12 and 2x ^(2) - kx - 9,` then find the value of k.

Options

A. 3
B. `-3`
C. `-4`
D. 4

Answer

To find the value of \( k \) given that \( (x - k) \) is the highest common factor (HCF) of the polynomials \( x^2 + x - 12 \) and \( 2x^2 - kx - 9 \), we can follow these steps:

### Step 1: Set \( x - k = 0 \)
Since \( (x - k) \) is a factor of both polynomials, we can set \( x = k \).

### Step 2: Substitute \( x = k \) into the first polynomial
Substituting \( x = k \) into the first polynomial \( x^2 + x - 12 \):
\[ k^2 + k - 12 = 0 \]

### Step 3: Substitute \( x = k \) into the second polynomial
Substituting \( x = k \) into the second polynomial \( 2x^2 - kx - 9 \):
\[ 2k^2 - k^2 - 9 = 0 \implies k^2 - 9 = 0 \]

### Step 4: Solve the equations
Now we have two equations:
1. \( k^2 + k - 12 = 0 \)
2. \( k^2 - 9 = 0 \)

From the second equation, we can factor it:
\[ (k - 3)(k + 3) = 0 \]
This gives us two possible values for \( k \):
\[ k = 3 \quad \text{or} \quad k = -3 \]

### Step 5: Substitute \( k = 3 \) into the first equation
Now we will check if \( k = 3 \) satisfies the first equation:
\[ 3^2 + 3 - 12 = 9 + 3 - 12 = 0 \]
This is true.

### Step 6: Substitute \( k = -3 \) into the first equation
Now we will check if \( k = -3 \) satisfies the first equation:
\[ (-3)^2 + (-3) - 12 = 9 - 3 - 12 = -6 \quad \text{(not true)} \]

### Conclusion
The only valid solution is:
\[ k = 3 \]

Thus, the value of \( k \) is \( 3 \).

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