The HCF of `a^2-ab-2b^2` and `2a^2-ab-b^2` is :

Options

A. (a+b)
B. (a-b)
C. (2a-3b)
D. none of these

Answer

To find the HCF (Highest Common Factor) of the two expressions \( f_1 = a^2 - ab - 2b^2 \) and \( f_2 = 2a^2 - ab - b^2 \), we will follow these steps:

### Step 1: Factor the first expression \( f_1 = a^2 - ab - 2b^2 \)

To factor \( f_1 \), we look for two numbers that multiply to \( -2b^2 \) (the product of the coefficient of \( a^2 \) and the constant term) and add to \( -b \) (the coefficient of \( ab \)).

The numbers that work are \( -2b \) and \( b \).

So, we can rewrite \( f_1 \) as:
\[ f_1 = a^2 - 2ab + ab - 2b^2 = (a^2 - 2ab) + (ab - 2b^2) \]
Now, we can factor by grouping:
\[ = a(a - 2b) + b(a - 2b) = (a - 2b)(a + b) \]

### Step 2: Factor the second expression \( f_2 = 2a^2 - ab - b^2 \)

For \( f_2 \), we look for two numbers that multiply to \( -2b^2 \) and add to \( -ab \).

The numbers that work are \( -b \) and \( 2b \).

So, we can rewrite \( f_2 \) as:
\[ f_2 = 2a^2 - 2ab + ab - b^2 = (2a^2 - 2ab) + (ab - b^2) \]
Now, we can factor by grouping:
\[ = 2a(a - b) + b(a - b) = (a - b)(2a + b) \]

### Step 3: Identify the common factors

Now we have:
\[ f_1 = (a - 2b)(a + b) \]
\[ f_2 = (a - b)(2a + b) \]

The common factors between \( f_1 \) and \( f_2 \) need to be identified.

- The factors of \( f_1 \) are \( (a - 2b) \) and \( (a + b) \).
- The factors of \( f_2 \) are \( (a - b) \) and \( (2a + b) \).

### Step 4: Determine the HCF

Looking at the factors, we see that there are no common factors between \( f_1 \) and \( f_2 \). Therefore, the HCF is:
\[ \text{HCF} = 1 \]

### Final Answer

The HCF of \( a^2 - ab - 2b^2 \) and \( 2a^2 - ab - b^2 \) is \( 1 \).

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