The HCF of two polynomials `4x^2(x^2 – 3x+2)` and `12x(x-2)(x^2 - 4)` is `4x(x-2)`. The LCM of the two polynomials is :

Options

A. `12x(x^2 - 4)`
B. `12x^2(x^2 – 3x+4) (x^2 - 2)`
C. `12x^2(x^2 – 3x+2) (x^2 - 4)`
D. `12x(x^2 – 3x-2)(x^2 - 4) `

Answer

To find the LCM of the two polynomials \(4x^2(x^2 - 3x + 2)\) and \(12x(x - 2)(x^2 - 4)\), we can use the relationship between the HCF (Highest Common Factor) and LCM (Lowest Common Multiple) of two polynomials:

\[ \text{LCM} \times \text{HCF} = P_1 \times P_2 \]

Where:
- \(P_1 = 4x^2(x^2 - 3x + 2)\)
- \(P_2 = 12x(x - 2)(x^2 - 4)\)
- HCF is given as \(4x(x - 2)\)

### Step 1: Calculate \(P_1 \times P_2\)
First, we need to compute the product of the two polynomials.

\[ P_1 \times P_2 = 4x^2(x^2 - 3x + 2) \times 12x(x - 2)(x^2 - 4) \]

### Step 2: Substitute the HCF into the LCM formula
We know that:

\[ \text{LCM} = \frac{P_1 \times P_2}{\text{HCF}} \]

Substituting the values we have:

\[ \text{LCM} = \frac{4x^2(x^2 - 3x + 2) \times 12x(x - 2)(x^2 - 4)}{4x(x - 2)} \]

### Step 3: Simplify the expression
Now we can simplify the expression by canceling out the common factors in the numerator and the denominator.

1. Cancel \(4\) from the numerator and denominator.
2. Cancel \(x\) from the numerator and denominator.
3. Cancel \((x - 2)\) from the numerator and denominator.

After canceling, we have:

\[ \text{LCM} = 12x(x^2 - 3x + 2)(x^2 - 4) \]

### Step 4: Factor the remaining polynomials
Next, we can factor the remaining polynomials:

1. \(x^2 - 3x + 2 = (x - 1)(x - 2)\)
2. \(x^2 - 4 = (x - 2)(x + 2)\)

Substituting these factored forms back into the LCM expression:

\[ \text{LCM} = 12x(x - 1)(x - 2)(x - 2)(x + 2) \]

### Step 5: Combine the factors
We can combine the factors to get the final LCM:

\[ \text{LCM} = 12x(x - 1)(x - 2)^2(x + 2) \]

### Final Answer
Thus, the LCM of the two polynomials is:

\[ \text{LCM} = 12x(x - 1)(x - 2)^2(x + 2) \]

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