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

Options

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

Answer

To find the HCF (Highest Common Factor) of the polynomials \(x^2 - y^2\) and \(x^3 - y^3\), we can follow these steps:

### Step 1: Factor the first polynomial \(x^2 - y^2\)
We can use the identity for the difference of squares:
\[ a^2 - b^2 = (a + b)(a - b) \]
Applying this to \(x^2 - y^2\):
\[ x^2 - y^2 = (x + y)(x - y) \]

### Step 2: Factor the second polynomial \(x^3 - y^3\)
We can use the identity for the difference of cubes:
\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]
Applying this to \(x^3 - y^3\):
\[ x^3 - y^3 = (x - y)(x^2 + xy + y^2) \]

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

The common factor between these two factorizations is:
\[ x - y \]

### Step 4: Conclusion
Thus, the HCF of the polynomials \(x^2 - y^2\) and \(x^3 - y^3\) is:
\[ \text{HCF} = x - y \]

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