The LCM of (x-1)(x-2) and `x^(2)(x-2)(x+3)` is

Options

A. (x-1)
B. (x-1)(x-2)(x+3)
C. `x^(2)(x-1)(x-2)(x+3)`
D. None of these

Answer

To find the LCM of the expressions \((x-1)(x-2)\) and \(x^2(x-2)(x+3)\), we will follow these steps:

### Step 1: Identify the factors of each expression
- The first expression is \((x-1)(x-2)\).
- The second expression is \(x^2(x-2)(x+3)\).

### Step 2: List the unique factors from both expressions
- From the first expression \((x-1)(x-2)\), the factors are:
- \(x-1\)
- \(x-2\)

- From the second expression \(x^2(x-2)(x+3)\), the factors are:
- \(x^2\) (which can be broken down into \(x\) and \(x\))
- \(x-2\)
- \(x+3\)

### Step 3: Combine the factors for the LCM
- The LCM must include each unique factor at its highest power:
- \(x-1\) (from the first expression)
- \(x-2\) (common in both, but we take it from either)
- \(x^2\) (from the second expression, which is higher than just \(x\))
- \(x+3\) (from the second expression)

### Step 4: Write the LCM
Combining all the factors, we get:
\[ \text{LCM} = x^2(x-1)(x-2)(x+3) \]

### Final Answer
Thus, the LCM of \((x-1)(x-2)\) and \(x^2(x-2)(x+3)\) is:
\[ x^2(x-1)(x-2)(x+3) \]

---