If the HCF of `x^3 - 343 ` and `x^2 - 9x + 14` is `(x - 7)` then find their LCM.

Answer

To find the LCM of the polynomials \(x^3 - 343\) and \(x^2 - 9x + 14\) given that their HCF is \(x - 7\), we can follow these steps:

### Step-by-Step Solution:

1. **Factor the first polynomial \(x^3 - 343\)**:
- Recognize that \(x^3 - 343\) can be expressed as a difference of cubes:
\[ x^3 - 7^3 = (x - 7)(x^2 + 7x + 49) \]
- Therefore, the factorization is:
\[ x^3 - 343 = (x - 7)(x^2 + 7x + 49) \]

2. **Factor the second polynomial \(x^2 - 9x + 14\)**:
- We can factor this quadratic by finding two numbers that multiply to \(14\) and add to \(-9\). The numbers are \(-2\) and \(-7\):
\[ x^2 - 9x + 14 = (x - 2)(x - 7) \]

3. **Write down the factorizations**:
- From the above steps, we have:
- \(x^3 - 343 = (x - 7)(x^2 + 7x + 49)\)
- \(x^2 - 9x + 14 = (x - 2)(x - 7)\)

4. **Use the relationship between HCF and LCM**:
- The relationship between HCF and LCM of two polynomials \(P_1\) and \(P_2\) is given by:
\[ \text{HCF}(P_1, P_2) \times \text{LCM}(P_1, P_2) = P_1 \times P_2 \]
- Here, \(P_1 = x^3 - 343\) and \(P_2 = x^2 - 9x + 14\). Thus:
\[ (x - 7) \times \text{LCM}(P_1, P_2) = (x^3 - 343)(x^2 - 9x + 14) \]

5. **Calculate the product of the two polynomials**:
- Substitute the factored forms:
\[ (x - 7)(x^2 + 7x + 49) \times (x - 2)(x - 7) \]
- This simplifies to:
\[ (x - 7)^2 (x^2 + 7x + 49)(x - 2) \]

6. **Solve for LCM**:
- Rearranging gives:
\[ \text{LCM}(P_1, P_2) = \frac{(x^3 - 343)(x^2 - 9x + 14)}{x - 7} \]
- Canceling \(x - 7\) from the numerator and denominator:
\[ \text{LCM}(P_1, P_2) = (x^2 + 7x + 49)(x - 2) \]

### Final Answer:
The LCM of the polynomials \(x^3 - 343\) and \(x^2 - 9x + 14\) is:
\[ \text{LCM} = (x^2 + 7x + 49)(x - 2) \]