HCF and LCM of two polynomials are (x+y) and `3x^(5) + 5x^(4)y + 2x^(3)y^(2) - 3x^(2)y^(3) - 5xy^(4) - 2y^(5)`, respectively. If one of the polynomials is `(x^(2) - y^(2))`. Then, the other polynomial is

Options

A. `3x^(4) + 8x^(3)y + 10x^(2)y^(2) + 2y^(4)`
B. `3x^(4) + 8x^(3)y + 10x^(2) y^(2) + 7xy^(3) + 2y^(4)`
C. `3x^(4) + 8x^(3)y + 10x^(2)y^(2) + 7xy^(3) + 2y^(4)`
D. None of the above

Answer

To find the other polynomial given the HCF and LCM of two polynomials, we can use the relationship:

**HCF × LCM = P1 × P2**

Where:
- HCF is the highest common factor of the two polynomials.
- LCM is the least common multiple of the two polynomials.
- P1 and P2 are the two polynomials.

Given:
- HCF = \(x + y\)
- LCM = \(3x^5 + 5x^4y + 2x^3y^2 - 3x^2y^3 - 5xy^4 - 2y^5\)
- One polynomial \(P1 = x^2 - y^2\)

We need to find the other polynomial \(P2\).

### Step-by-Step Solution:

1. **Write the relationship:**
\[ (x + y) \times \left(3x^5 + 5x^4y + 2x^3y^2 - 3x^2y^3 - 5xy^4 - 2y^5\right) = (x^2 - y^2) \times P2 \]

2. **Factor \(x^2 - y^2\):**
\[ x^2 - y^2 = (x + y)(x - y) \]
Thus, we can rewrite the equation as:
\[ (x + y) \times \left(3x^5 + 5x^4y + 2x^3y^2 - 3x^2y^3 - 5xy^4 - 2y^5\right) = (x + y)(x - y) \times P2 \]

3. **Cancel \(x + y\) from both sides (assuming \(x + y \neq 0\)):**
\[ 3x^5 + 5x^4y + 2x^3y^2 - 3x^2y^3 - 5xy^4 - 2y^5 = (x - y) \times P2 \]

4. **Divide both sides by \(x - y\) to isolate \(P2\):**
\[ P2 = \frac{3x^5 + 5x^4y + 2x^3y^2 - 3x^2y^3 - 5xy^4 - 2y^5}{x - y} \]

5. **Perform polynomial long division:**
- Divide \(3x^5\) by \(x - y\) to find the first term of \(P2\).
- Continue the division process until all terms are accounted for.

6. **After performing the division, we find:**
\[ P2 = 3x^4 + 8x^3y + 10x^2y^2 + 7xy^3 + 2y^4 \]

### Final Answer:
The other polynomial \(P2\) is:
\[ P2 = 3x^4 + 8x^3y + 10x^2y^2 + 7xy^3 + 2y^4 \]