Find the LCM of the polynomials `5(x^(3)-y^(3)) and 35(x^(6)-y^(6))`.

Options

A. `(x-y)`
B. `5(x^(3)-y^(3))`
C. `35(x^(3)-y^(3))`
D. `35(x^(6)-y^(6))`

Answer

To find the LCM of the polynomials \(5(x^3 - y^3)\) and \(35(x^6 - y^6)\), we will follow these steps:

### Step 1: Factor the polynomials
First, we need to factor both polynomials.

1. **Factor \(5(x^3 - y^3)\)**:
- We can use the identity for the difference of cubes:
\[ x^3 - y^3 = (x - y)(x^2 + xy + y^2) \]
- Therefore, we can write:
\[ 5(x^3 - y^3) = 5(x - y)(x^2 + xy + y^2) \]

2. **Factor \(35(x^6 - y^6)\)**:
- We can use the identity for the difference of squares:
\[ x^6 - y^6 = (x^3 - y^3)(x^3 + y^3) \]
- We already factored \(x^3 - y^3\) in the previous step, and we can factor \(x^3 + y^3\) using the sum of cubes:
\[ x^3 + y^3 = (x + y)(x^2 - xy + y^2) \]
- Thus, we have:
\[ x^6 - y^6 = (x - y)(x^2 + xy + y^2)(x + y)(x^2 - xy + y^2) \]
- Therefore, we can write:
\[ 35(x^6 - y^6) = 35(x - y)(x^2 + xy + y^2)(x + y)(x^2 - xy + y^2) \]

### Step 2: Identify the LCM
Now that we have factored both polynomials, we can identify the LCM.

- The LCM is found by taking the highest power of each factor that appears in the factorizations.

1. **Factors from \(5(x^3 - y^3)\)**:
- \(5\)
- \((x - y)\)
- \((x^2 + xy + y^2)\)

2. **Factors from \(35(x^6 - y^6)\)**:
- \(35 = 7 \times 5\)
- \((x - y)\)
- \((x^2 + xy + y^2)\)
- \((x + y)\)
- \((x^2 - xy + y^2)\)

### Step 3: Combine the factors for LCM
- The LCM will be:
- The highest coefficient: \(35\)
- The highest powers of each factor:
- \(x - y\) (appears in both)
- \(x^2 + xy + y^2\) (appears in both)
- \(x + y\) (only in the second polynomial)
- \(x^2 - xy + y^2\) (only in the second polynomial)

Thus, the LCM is:
\[ \text{LCM} = 35(x - y)(x^2 + xy + y^2)(x + y)(x^2 - xy + y^2) \]

### Final Answer
The LCM of the polynomials \(5(x^3 - y^3)\) and \(35(x^6 - y^6)\) is:
\[ 35(x - y)(x^2 + xy + y^2)(x + y)(x^2 - xy + y^2) \]

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