The LCM of `6(x^2 + xy), 8(xy - y^2), 12 (x^2 -y^2)` and `20(x + y)^2` is:

Options

A. `120 x (x + y)(x-y)`
B. `120 xy(x + y)(x-y)`
C. `120 xy(x + y)^2 (x - y)`
D. `120xy (x + y)(x - y)^2`

Answer

To find the LCM of the given polynomials \(6(x^2 + xy)\), \(8(xy - y^2)\), \(12(x^2 - y^2)\), and \(20(x + y)^2\), we will follow these steps:

### Step 1: Factor each polynomial
1. **Factor \(6(x^2 + xy)\)**:
\[ 6(x^2 + xy) = 6x(x + y) = 2 \cdot 3 \cdot x(x + y) \]

2. **Factor \(8(xy - y^2)\)**:
\[ 8(xy - y^2) = 8y(x - y) = 2^3 \cdot y(x - y) \]

3. **Factor \(12(x^2 - y^2)\)**:
\[ 12(x^2 - y^2) = 12(x + y)(x - y) = 2^2 \cdot 3 \cdot (x + y)(x - y) \]

4. **Factor \(20(x + y)^2\)**:
\[ 20(x + y)^2 = 20(x + y)(x + y) = 2^2 \cdot 5 \cdot (x + y)^2 \]

### Step 2: Identify the highest power of each factor
Now, we will identify the highest power of each prime factor and polynomial factor from the factorizations:

- For the factor \(2\):
- From \(6\): \(2^1\)
- From \(8\): \(2^3\)
- From \(12\): \(2^2\)
- From \(20\): \(2^2\)
- **Maximum**: \(2^3\)

- For the factor \(3\):
- From \(6\): \(3^1\)
- From \(8\): \(3^0\)
- From \(12\): \(3^1\)
- From \(20\): \(3^0\)
- **Maximum**: \(3^1\)

- For the factor \(5\):
- From \(6\): \(5^0\)
- From \(8\): \(5^0\)
- From \(12\): \(5^0\)
- From \(20\): \(5^1\)
- **Maximum**: \(5^1\)

- For the polynomial factor \((x + y)\):
- From \(6\): \((x + y)^1\)
- From \(8\): \((x + y)^0\)
- From \(12\): \((x + y)^1\)
- From \(20\): \((x + y)^2\)
- **Maximum**: \((x + y)^2\)

- For the polynomial factor \((x - y)\):
- From \(6\): \((x - y)^0\)
- From \(8\): \((x - y)^1\)
- From \(12\): \((x - y)^1\)
- From \(20\): \((x - y)^0\)
- **Maximum**: \((x - y)^1\)

- For the polynomial factor \(x\):
- From \(6\): \(x^1\)
- From \(8\): \(x^0\)
- From \(12\): \(x^2\)
- From \(20\): \(x^0\)
- **Maximum**: \(x^2\)

- For the polynomial factor \(y\):
- From \(6\): \(y^0\)
- From \(8\): \(y^1\)
- From \(12\): \(y^0\)
- From \(20\): \(y^0\)
- **Maximum**: \(y^1\)

### Step 3: Combine the factors to find the LCM
Now we can write the LCM by multiplying the highest powers of all the factors:

\[ \text{LCM} = 2^3 \cdot 3^1 \cdot 5^1 \cdot (x + y)^2 \cdot (x - y)^1 \cdot x^2 \cdot y^1 \]

Calculating the numerical part:
\[ 2^3 = 8, \quad 3^1 = 3, \quad 5^1 = 5 \]
\[ 8 \cdot 3 \cdot 5 = 120 \]

Thus, the LCM is:
\[ \text{LCM} = 120 \cdot x^2 \cdot y \cdot (x + y)^2 \cdot (x - y) \]

### Final Answer:
\[ \text{LCM} = 120 \cdot (x + y)^2 \cdot (x - y) \cdot x^2 \cdot y \]