What is the LCM of 3(a3 - b3) and 11(a4 - b4)?

To find the least common multiple (LCM) of \( 3(a^3 - b^3) \) and \( 11(a^4 - b^4) \), we first factor each expression.

### Step 1: Factor the Terms

1. **For \( a^3 - b^3 \):**
   \[
   a^3 - b^3 = (a - b)(a^2 + ab + b^2)
   \]
   Thus, \( 3(a^3 - b^3) = 3(a - b)(a^2 + ab + b^2) \).

2. **For \( a^4 - b^4 \):**
   This is a difference of squares, which factors as:
   \[
   a^4 - b^4 = (a^2 - b^2)(a^2 + b^2)
   \]
   and further, \( a^2 - b^2 \) can be factored as \( (a - b)(a + b) \). So,
   \[
   a^4 - b^4 = (a - b)(a + b)(a^2 + b^2)
   \]
   Therefore,
   \[
   11(a^4 - b^4) = 11(a - b)(a + b)(a^2 + b^2)
   \]

### Step 2: Identify the LCM Components
Now, the LCM must include each unique factor to the highest power in which it appears across both expressions:

- The terms \( 3 \) and \( 11 \) appear individually, so the LCM will contain \( 3 \cdot 11 = 33 \).
- The factor \( (a - b) \) appears in both expressions to the first power.
- The factor \( (a + b) \) only appears in \( 11(a^4 - b^4) \).
- The factor \( (a^2 + ab + b^2) \) appears only in \( 3(a^3 - b^3) \).
- The factor \( (a^2 + b^2) \) appears only in \( 11(a^4 - b^4) \).

### Step 3: Write the LCM
Combining all factors at their highest powers, the LCM is:
\[
\text{LCM} = 33(a - b)(a + b)(a^2 + ab + b^2)(a^2 + b^2)
\]

### Final Answer
\[
\text{LCM of } 3(a^3 - b^3) \text{ and } 11(a^4 - b^4) = 33(a - b)(a + b)(a^2 + ab + b^2)(a^2 + b^2)
\]

Would you like more details or have any questions?

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1. How do we determine the greatest common divisor (GCD) in polynomial expressions?
2. What are the general rules for finding the LCM of polynomial expressions?
3. Why do we factor out terms before calculating the LCM in polynomial expressions?
4. How do differences of cubes and squares factor, and why are these patterns useful?
5. How do we apply the LCM concept to rational expressions?

**Tip:** For finding the LCM of polynomials, factor each completely and use each factor at its highest power across all expressions.

What is the LCM of 3(a^3 - b^3) and 11(a^4 - b^4)?

Explore how to determine the least common multiple (LCM) of the polynomials 3(a^3 - b^3) and 11(a^4 - b^4). This solution demonstrates factorization techniques for polynomials and explains the process of identifying the LCM by using each factor at its highest power. Ideal for students in Grades 10-12.

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