HFC of ` x ^(2) - y ^(2) and x ^(3) - y ^(3)` is -

Options

A. `x-y`
B. `x ^(3) - y ^(3)`
C. `(x ^(2) - y ^(2))`
D. `(x +y), (x ^(2) + xy + y ^(2))`

Answer

To find the highest common factor (HCF) of the expressions \( x^2 - y^2 \) and \( x^3 - y^3 \), we can follow these steps:

### Step 1: Factor \( x^2 - y^2 \)
The expression \( x^2 - y^2 \) can be factored using the difference of squares formula:
\[ x^2 - y^2 = (x - y)(x + y) \]

### Step 2: Factor \( x^3 - y^3 \)
The expression \( x^3 - y^3 \) can be factored using the difference of cubes formula:
\[ x^3 - y^3 = (x - y)(x^2 + xy + y^2) \]

### Step 3: Identify the common factors
Now we have the factored forms:
- From \( x^2 - y^2 \): \( (x - y)(x + y) \)
- From \( x^3 - y^3 \): \( (x - y)(x^2 + xy + y^2) \)

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

### Step 4: Determine the HCF
Since \( (x - y) \) is the only common factor, we conclude that the highest common factor (HCF) of \( x^2 - y^2 \) and \( x^3 - y^3 \) is:
\[ \text{HCF} = x - y \]

### Final Answer
Thus, the HCF of \( x^2 - y^2 \) and \( x^3 - y^3 \) is \( x - y \).

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