The HCF of `x^(2) - xy - 2y^(2)` and `2x^(2) - xy - y^(2)` is

Options

A. (x-y)
B. (x+y)
C. (2x-3y)
D. None of these

Answer

To find the HCF of the polynomials \( x^{2} - xy - 2y^{2} \) and \( 2x^{2} - xy - y^{2} \), we will follow these steps:

### Step 1: Factor the first polynomial \( x^{2} - xy - 2y^{2} \)

We need to factor the polynomial \( x^{2} - xy - 2y^{2} \).

To do this, we look for two numbers that multiply to \( -2y^{2} \) (the product of the coefficient of \( x^{2} \) and the constant term) and add up to \( -y \) (the coefficient of \( xy \)).

The numbers that satisfy this are \( -2y \) and \( y \).

So, we can rewrite the polynomial as:
\[ x^{2} - 2xy + xy - 2y^{2} \]

Now, we can group the terms:
\[ (x^{2} - 2xy) + (xy - 2y^{2}) \]

Factoring out the common terms in each group:
\[ x(x - 2y) + y(x - 2y) \]

Now, we can factor out \( (x - 2y) \):
\[ (x - 2y)(x + y) \]

### Step 2: Factor the second polynomial \( 2x^{2} - xy - y^{2} \)

Next, we will factor the polynomial \( 2x^{2} - xy - y^{2} \).

We look for two numbers that multiply to \( 2(-y^{2}) = -2y^{2} \) and add up to \( -y \).

The numbers that satisfy this are \( -2y \) and \( y \).

So, we can rewrite the polynomial as:
\[ 2x^{2} - 2xy + xy - y^{2} \]

Now, we can group the terms:
\[ (2x^{2} - 2xy) + (xy - y^{2}) \]

Factoring out the common terms in each group:
\[ 2x(x - y) + y(x - y) \]

Now, we can factor out \( (x - y) \):
\[ (x - y)(2x + y) \]

### Step 3: Find the HCF of the two factored forms

Now we have:
1. \( x^{2} - xy - 2y^{2} = (x - 2y)(x + y) \)
2. \( 2x^{2} - xy - y^{2} = (x - y)(2x + y) \)

To find the HCF, we look for common factors in both factored forms.

The common factor is \( (x - y) \).

### Conclusion

Thus, the HCF of \( x^{2} - xy - 2y^{2} \) and \( 2x^{2} - xy - y^{2} \) is:
\[ \text{HCF} = (x - y) \]