Find the HCF and LCM of the following polynomials `(x+2)^(2)(x-3)^(2)(x+1)^(2),(x+1)^(3)(x+2)^(3)(x-3)`

Answer

To find the HCF (Highest Common Factor) and LCM (Least Common Multiple) of the given polynomials, we will follow these steps:

### Given Polynomials:
1. \( P_1 = (x + 2)^2 (x - 3)^2 (x + 1)^2 \)
2. \( P_2 = (x + 1)^3 (x + 2)^3 (x - 3) \)

### Step 1: Finding the HCF
The HCF is found by taking the common factors of the polynomials and selecting the minimum power for each common factor.

- **Common Factors:**
- For \( (x + 2) \):
- Power in \( P_1 = 2 \)
- Power in \( P_2 = 3 \)
- Minimum power = \( 2 \)
- Contribution to HCF: \( (x + 2)^2 \)

- For \( (x - 3) \):
- Power in \( P_1 = 2 \)
- Power in \( P_2 = 1 \)
- Minimum power = \( 1 \)
- Contribution to HCF: \( (x - 3)^1 \)

- For \( (x + 1) \):
- Power in \( P_1 = 2 \)
- Power in \( P_2 = 3 \)
- Minimum power = \( 2 \)
- Contribution to HCF: \( (x + 1)^2 \)

- **HCF Calculation:**
\[ \text{HCF} = (x + 2)^2 (x - 3)^1 (x + 1)^2 \]

### Step 2: Finding the LCM
The LCM is found by taking the common factors and selecting the maximum power for each factor.

- **Common Factors:**
- For \( (x + 2) \):
- Power in \( P_1 = 2 \)
- Power in \( P_2 = 3 \)
- Maximum power = \( 3 \)
- Contribution to LCM: \( (x + 2)^3 \)

- For \( (x - 3) \):
- Power in \( P_1 = 2 \)
- Power in \( P_2 = 1 \)
- Maximum power = \( 2 \)
- Contribution to LCM: \( (x - 3)^2 \)

- For \( (x + 1) \):
- Power in \( P_1 = 2 \)
- Power in \( P_2 = 3 \)
- Maximum power = \( 3 \)
- Contribution to LCM: \( (x + 1)^3 \)

- **LCM Calculation:**
\[ \text{LCM} = (x + 2)^3 (x - 3)^2 (x + 1)^3 \]

### Final Results:
- **HCF:** \( (x + 2)^2 (x - 3) (x + 1)^2 \)
- **LCM:** \( (x + 2)^3 (x - 3)^2 (x + 1)^3 \)