What is the LCM of `(6x^(3) +60x^(2) +150x) and (3x^(4) +12 x^(3) -15 x^(2))`?

Options

A. `6x^(2) (x+5)^(2) (x-1)`
B. `3x^(2) (x+5)^(2) (x-1)`
C. `6x^(2) (x+5)^(2) (x-1)^(2)`
D. `3x^(2) (x+5) (x-1)^(2)`

Answer

To find the LCM of the two polynomials \(6x^3 + 60x^2 + 150x\) and \(3x^4 + 12x^3 - 15x^2\), we will follow these steps:

### Step 1: Factor the first polynomial \(6x^3 + 60x^2 + 150x\)

1. **Identify the common factor**: The common factor in all the terms is \(6x\).
2. **Factor out \(6x\)**:
\[ 6x^3 + 60x^2 + 150x = 6x(x^2 + 10x + 25) \]
3. **Factor the quadratic \(x^2 + 10x + 25\)**:
\[ x^2 + 10x + 25 = (x + 5)^2 \]
4. **So, the complete factorization of the first polynomial is**:
\[ 6x^3 + 60x^2 + 150x = 6x(x + 5)^2 \]

### Step 2: Factor the second polynomial \(3x^4 + 12x^3 - 15x^2\)

1. **Identify the common factor**: The common factor in all the terms is \(3x^2\).
2. **Factor out \(3x^2\)**:
\[ 3x^4 + 12x^3 - 15x^2 = 3x^2(x^2 + 4x - 5) \]
3. **Factor the quadratic \(x^2 + 4x - 5\)**:
\[ x^2 + 4x - 5 = (x + 5)(x - 1) \]
4. **So, the complete factorization of the second polynomial is**:
\[ 3x^4 + 12x^3 - 15x^2 = 3x^2(x + 5)(x - 1) \]

### Step 3: Find the LCM

To find the LCM of the two polynomials, we take the highest power of each factor present in the factorizations:

1. From \(6x(x + 5)^2\), we have:
- \(6\)
- \(x\)
- \((x + 5)^2\)

2. From \(3x^2(x + 5)(x - 1)\), we have:
- \(3\)
- \(x^2\)
- \((x + 5)\)
- \((x - 1)\)

### Combine the factors:

- The LCM will be:
\[ \text{LCM} = \text{LCM of coefficients} \cdot \text{highest power of } x \cdot (x + 5)^2 \cdot (x - 1) \]

- **LCM of coefficients**: The LCM of \(6\) and \(3\) is \(6\).
- **Highest power of \(x\)**: The highest power is \(x^2\).
- **Combine the factors**:
\[ \text{LCM} = 6x^2(x + 5)^2(x - 1) \]

### Final Answer:
\[ \text{LCM} = 6x^2(x + 5)^2(x - 1) \]