Find the **LCM** of the polynomials:

$$x^2 - 8x + 15 \quad ext{and} \quad x^2 - 5x + 6$$

Question

Find the **LCM** of the polynomials:

$$x^2 - 8x + 15 \quad 	ext{and} \quad x^2 - 5x + 6$$

Accepted Answer

**Solution:**

1. **Factor the first polynomial:**
   - Given: $x^2 - 8x + 15$
   - Find two numbers that multiply to $15$ and add to $-8$: these are $-3$ and $-5$.
   - So, $x^2 - 8x + 15 = (x - 3)(x - 5)$.

2. **Factor the second polynomial:**
   - Given: $x^2 - 5x + 6$
   - Find two numbers that multiply to $6$ and add to $-5$: these are $-2$ and $-3$.
   - So, $x^2 - 5x + 6 = (x - 2)(x - 3)$.

3. **Find the LCM:**
   - The LCM is the product of the highest powers of all distinct factors from both polynomials.
   - Distinct factors: $(x - 2)$, $(x - 3)$, $(x - 5)$.
   - Each factor appears to the power 1 in the factorizations.
   - Therefore, the LCM is:
     $$	ext{LCM} = (x - 2)(x - 3)(x - 5)$$

**Answer:** $(x - 2)(x - 3)(x - 5)$

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