The LCM of `x^2 – 3x +2` and `x^3 – 2x^2 – 3x` is :

Options

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

Answer

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

### Step 1: Factor the first polynomial \(x^2 - 3x + 2\)
To factor \(x^2 - 3x + 2\), we look for two numbers that multiply to \(2\) (the constant term) and add up to \(-3\) (the coefficient of \(x\)). The numbers \(-1\) and \(-2\) satisfy this condition.

Thus, we can factor it as:
\[ x^2 - 3x + 2 = (x - 1)(x - 2) \]

### Step 2: Factor the second polynomial \(x^3 - 2x^2 - 3x\)
First, we can factor out \(x\) from the polynomial:
\[ x^3 - 2x^2 - 3x = x(x^2 - 2x - 3) \]
Next, we need to factor \(x^2 - 2x - 3\). We look for two numbers that multiply to \(-3\) and add to \(-2\). The numbers \(-3\) and \(1\) work.

Thus, we can factor it as:
\[ x^2 - 2x - 3 = (x - 3)(x + 1) \]
So, the complete factorization of the second polynomial is:
\[ x^3 - 2x^2 - 3x = x(x - 3)(x + 1) \]

### Step 3: Identify the factors from both polynomials
Now we have:
- From \(x^2 - 3x + 2\): \((x - 1)(x - 2)\)
- From \(x^3 - 2x^2 - 3x\): \(x(x - 3)(x + 1)\)

### Step 4: Determine the LCM
The LCM is found by taking the highest power of each factor that appears in the factorizations:
- \(x\) from \(x^3 - 2x^2 - 3x\)
- \(x - 1\) from \(x^2 - 3x + 2\)
- \(x - 2\) from \(x^2 - 3x + 2\)
- \(x - 3\) from \(x^3 - 2x^2 - 3x\)
- \(x + 1\) from \(x^3 - 2x^2 - 3x\)

Thus, the LCM is:
\[ \text{LCM} = x(x - 1)(x - 2)(x - 3)(x + 1) \]

### Step 5: Simplify the LCM expression
Notice that \((x - 1)(x + 1) = x^2 - 1\), so we can rewrite the LCM as:
\[ \text{LCM} = x(x - 2)(x - 3)(x^2 - 1) \]

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

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