Find the LCM and HCF of the polynomials `15x^(2) y^(3) z, 3x^(3) yz^(2)`

Answer

To find the LCM (Least Common Multiple) and HCF (Highest Common Factor) of the polynomials \(15x^2y^3z\) and \(3x^3yz^2\), we will follow these steps:

### Step 1: Factor the polynomials
1. **Factor \(15x^2y^3z\)**:
- The coefficient \(15\) can be factored as \(3 \times 5\).
- The variable \(x^2\) can be expressed as \(x \times x\).
- The variable \(y^3\) can be expressed as \(y \times y \times y\).
- The variable \(z\) remains \(z\).
- Therefore, \(15x^2y^3z = 3 \times 5 \times x \times x \times y \times y \times y \times z\).

2. **Factor \(3x^3yz^2\)**:
- The coefficient \(3\) remains \(3\).
- The variable \(x^3\) can be expressed as \(x \times x \times x\).
- The variable \(y\) remains \(y\).
- The variable \(z^2\) can be expressed as \(z \times z\).
- Therefore, \(3x^3yz^2 = 3 \times x \times x \times x \times y \times z \times z\).

### Step 2: Find the LCM
3. **Identify the highest powers of each factor**:
- Coefficient: The highest power of \(3\) is \(3\) (common in both), and \(5\) appears only in the first polynomial.
- For \(x\): The highest power is \(x^3\) from the second polynomial.
- For \(y\): The highest power is \(y^3\) from the first polynomial.
- For \(z\): The highest power is \(z^2\) from the second polynomial.

4. **Construct the LCM**:
- LCM = \(3 \times 5 \times x^3 \times y^3 \times z^2\)
- LCM = \(15x^3y^3z^2\)

### Step 3: Find the HCF
5. **Identify the lowest powers of each common factor**:
- Coefficient: The common factor is \(3\).
- For \(x\): The lowest power is \(x^2\) from the first polynomial.
- For \(y\): The lowest power is \(y^1\) from the second polynomial.
- For \(z\): The lowest power is \(z^1\) from the first polynomial.

6. **Construct the HCF**:
- HCF = \(3 \times x^2 \times y^1 \times z^1\)
- HCF = \(3x^2yz\)

### Final Answer
- **LCM**: \(15x^3y^3z^2\)
- **HCF**: \(3x^2yz\)