The LCM of `2(a^(2)-b^(2)), 3(a^(3) - b^(3)), 4(a^(4) - b^(4))` is

Options

A. `6(a-b)(a+b)(a^(2) + b^(2))`
B. `12(a^(4)-b^(4)) (a^(2) + ab + b^(2))`
C. `a^(3) - b^(3)`
D. `12(a^(4) - b^(4))`

Answer

To find the LCM of the expressions \(2(a^2 - b^2)\), \(3(a^3 - b^3)\), and \(4(a^4 - b^4)\), we will follow these steps:

### Step 1: Factor each expression

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

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

3. **Factor \(4(a^4 - b^4)\)**:
\[ a^4 - b^4 = (a^2 - b^2)(a^2 + b^2) = (a + b)(a - b)(a^2 + b^2) \]
Thus,
\[ 4(a^4 - b^4) = 4(a + b)(a - b)(a^2 + b^2) \]

### Step 2: Identify the LCM

Now we have:
- \(2(a + b)(a - b)\)
- \(3(a - b)(a^2 + ab + b^2)\)
- \(4(a + b)(a - b)(a^2 + b^2)\)

To find the LCM, we take the highest power of each factor present in any of the expressions:

- The highest coefficient is \(4\) (from \(4(a^4 - b^4)\)).
- The factors are:
- \(a + b\) (from \(4(a + b)(a - b)(a^2 + b^2)\))
- \(a - b\) (common in all three)
- \(a^2 + ab + b^2\) (from \(3(a^3 - b^3)\))
- \(a^2 + b^2\) (from \(4(a^4 - b^4)\))

### Step 3: Combine the factors

Thus, the LCM is:
\[ LCM = 4(a + b)(a - b)(a^2 + ab + b^2)(a^2 + b^2) \]

### Final Result

The LCM of \(2(a^2 - b^2)\), \(3(a^3 - b^3)\), and \(4(a^4 - b^4)\) is:
\[ 4(a + b)(a - b)(a^2 + ab + b^2)(a^2 + b^2) \]

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