The LCM of the polynomials `(x+3)^(2)(x-2)(x+1)^(2)` and `(x+1)^(3)(x+3)(x+4)` is

Options

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

Answer

To find the LCM of the given polynomials \( (x+3)^{2}(x-2)(x+1)^{2} \) and \( (x+1)^{3}(x+3)(x+4) \), we will follow these steps:

### Step 1: Identify the factors and their powers in both polynomials.

For the first polynomial \( P = (x+3)^{2}(x-2)(x+1)^{2} \):
- \( (x+3) \) has a power of 2.
- \( (x-2) \) has a power of 1.
- \( (x+1) \) has a power of 2.

For the second polynomial \( Q = (x+1)^{3}(x+3)(x+4) \):
- \( (x+1) \) has a power of 3.
- \( (x+3) \) has a power of 1.
- \( (x+4) \) has a power of 1.

### Step 2: Determine the highest power of each factor.

Now we will take the highest power of each distinct factor from both polynomials:
- For \( (x+3) \): The highest power is \( \max(2, 1) = 2 \).
- For \( (x-2) \): The highest power is \( \max(1, 0) = 1 \) (since it only appears in \( P \)).
- For \( (x+1) \): The highest power is \( \max(2, 3) = 3 \).
- For \( (x+4) \): The highest power is \( \max(0, 1) = 1 \) (since it only appears in \( Q \)).

### Step 3: Write the LCM using the highest powers.

Now we can write the LCM of the two polynomials:
\[ \text{LCM} = (x+3)^{2} \cdot (x-2)^{1} \cdot (x+1)^{3} \cdot (x+4)^{1} \]

### Step 4: Simplify the expression.

Thus, the LCM can be expressed as:
\[ \text{LCM} = (x+3)^{2} (x-2) (x+1)^{3} (x+4) \]

### Final Answer:

The LCM of the polynomials \( (x+3)^{2}(x-2)(x+1)^{2} \) and \( (x+1)^{3}(x+3)(x+4) \) is:
\[ (x+3)^{2} (x-2) (x+1)^{3} (x+4) \]

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