The LCM of the polynomials `x^3 +3x^2 +3x + 1, x^2 +2x+1` and `x^2 -1` is :

Options

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

Answer

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

### Step 1: Factor each polynomial

1. **Factor \(x^3 + 3x^2 + 3x + 1\)**:
- This polynomial can be recognized as a perfect cube. It can be factored using the formula for the expansion of \((a + b)^3\).
- Here, \(a = x\) and \(b = 1\), so:
\[ x^3 + 3x^2 + 3x + 1 = (x + 1)^3 \]

2. **Factor \(x^2 + 2x + 1\)**:
- This is a perfect square trinomial. It can be factored as:
\[ x^2 + 2x + 1 = (x + 1)^2 \]

3. **Factor \(x^2 - 1\)**:
- This is a difference of squares and can be factored as:
\[ x^2 - 1 = (x - 1)(x + 1) \]

### Step 2: List the unique factors and their highest powers

Now we have the factorizations:
- \(x^3 + 3x^2 + 3x + 1 = (x + 1)^3\)
- \(x^2 + 2x + 1 = (x + 1)^2\)
- \(x^2 - 1 = (x - 1)(x + 1)\)

The unique factors are:
- \(x + 1\)
- \(x - 1\)

Now we will take the highest power of each factor:
- For \(x + 1\): The highest power is \(3\) (from \((x + 1)^3\)).
- For \(x - 1\): The highest power is \(1\) (from \((x - 1)\)).

### Step 3: Write the LCM

The LCM is obtained by multiplying the highest powers of all unique factors:
\[ \text{LCM} = (x + 1)^3 (x - 1)^1 \]

Thus, the LCM of the given polynomials is:
\[ \text{LCM} = (x + 1)^3 (x - 1) \]

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

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