The LCM of `x^(3) - 2x^(2) - x + 2` and `x^(3) - x^(2) - 4x + 4` is

Options

A. `(x-1)(x^(2) - 4)`
B. `(x^(2) - 1)(x^(2) - 4)`
C. `(x^(3) - 1)(x^(3) - 4)`
D. `(x - 1)(x - 4)`

Answer

To find the LCM of the polynomials \( P_1(x) = x^3 - 2x^2 - x + 2 \) and \( P_2(x) = x^3 - x^2 - 4x + 4 \), we will follow these steps:

### Step 1: Factor the first polynomial \( P_1(x) \)

1. **Identify the polynomial**:
\[ P_1(x) = x^3 - 2x^2 - x + 2 \]

2. **Group the terms**:
\[ = (x^3 - 2x^2) + (-x + 2) \]

3. **Factor out common terms**:
\[ = x^2(x - 2) - 1(x - 2) \]

4. **Factor by grouping**:
\[ = (x - 2)(x^2 - 1) \]

5. **Factor \( x^2 - 1 \)**:
\[ = (x - 2)(x - 1)(x + 1) \]

So,
\[ P_1(x) = (x - 2)(x - 1)(x + 1) \]

### Step 2: Factor the second polynomial \( P_2(x) \)

1. **Identify the polynomial**:
\[ P_2(x) = x^3 - x^2 - 4x + 4 \]

2. **Group the terms**:
\[ = (x^3 - x^2) + (-4x + 4) \]

3. **Factor out common terms**:
\[ = x^2(x - 1) - 4(x - 1) \]

4. **Factor by grouping**:
\[ = (x - 1)(x^2 - 4) \]

5. **Factor \( x^2 - 4 \)**:
\[ = (x - 1)(x - 2)(x + 2) \]

So,
\[ P_2(x) = (x - 1)(x - 2)(x + 2) \]

### Step 3: Find the LCM of \( P_1(x) \) and \( P_2(x) \)

1. **List the factors**:
- From \( P_1(x) \): \( (x - 2), (x - 1), (x + 1) \)
- From \( P_2(x) \): \( (x - 1), (x - 2), (x + 2) \)

2. **Identify the unique factors**:
- \( (x - 2) \)
- \( (x - 1) \)
- \( (x + 1) \)
- \( (x + 2) \)

3. **Combine the unique factors for the LCM**:
\[ \text{LCM}(P_1, P_2) = (x - 2)(x - 1)(x + 1)(x + 2) \]

### Final Answer:
\[ \text{LCM}(P_1(x), P_2(x)) = (x - 2)(x - 1)(x + 1)(x + 2) \]

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