If (x-a) is the HCF of `x^(2) - x - 6` and `x^(2) + 3x - 18`. Then, the value of a is

Options

A. 3
B. 4
C. 1
D. 2

Answer

To find the value of \( a \) such that \( (x - a) \) is the HCF of the polynomials \( x^2 - x - 6 \) and \( x^2 + 3x - 18 \), we can follow these steps:

### Step 1: Factor the first polynomial \( x^2 - x - 6 \)

To factor \( x^2 - x - 6 \), we need to find two numbers that multiply to \(-6\) (the constant term) and add to \(-1\) (the coefficient of \( x \)).

The numbers \(-3\) and \(2\) satisfy these conditions:

\[ x^2 - x - 6 = (x - 3)(x + 2) \]

### Step 2: Factor the second polynomial \( x^2 + 3x - 18 \)

Next, we factor \( x^2 + 3x - 18 \). We need two numbers that multiply to \(-18\) and add to \(3\).

The numbers \(6\) and \(-3\) satisfy these conditions:

\[ x^2 + 3x - 18 = (x + 6)(x - 3) \]

### Step 3: Identify the HCF

Now we have the factored forms of both polynomials:

1. \( x^2 - x - 6 = (x - 3)(x + 2) \)
2. \( x^2 + 3x - 18 = (x + 6)(x - 3) \)

The common factor in both expressions is \( (x - 3) \).

### Step 4: Relate the HCF to \( x - a \)

Since we are given that \( (x - a) \) is the HCF of the two polynomials, we can set:

\[ x - a = x - 3 \]

### Step 5: Solve for \( a \)

From the equation \( x - a = x - 3 \), we can equate the constants:

\[ -a = -3 \implies a = 3 \]

### Final Answer

Thus, the value of \( a \) is:

\[ \boxed{3} \]

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