The LCM of the polynomials `(x^(8)-y^(8))` and `(x^(4)-y^(4))(x+y)` is

Options

A. `(x^(8)-y^(8))`
B. `(x^(8)-y^(8))(x^(4)-y^(4))(x+y)`
C. `(x^(8)-y^(8))(x+y)`
D. `(x^(4)-y^(4))(x^(8)-y^(8))`

Answer

To find the LCM of the polynomials \(x^8 - y^8\) and \((x^4 - y^4)(x + y)\), we can follow these steps:

### Step 1: Factor the first polynomial \(x^8 - y^8\)

We can use the difference of squares identity:
\[ x^8 - y^8 = (x^4)^2 - (y^4)^2 = (x^4 - y^4)(x^4 + y^4) \]

### Step 2: Write the second polynomial

The second polynomial is already given as:
\[ (x^4 - y^4)(x + y) \]

### Step 3: Identify the common factors

Now we have:
1. From the first polynomial: \( (x^4 - y^4)(x^4 + y^4) \)
2. From the second polynomial: \( (x^4 - y^4)(x + y) \)

The common factor between both polynomials is \(x^4 - y^4\).

### Step 4: Determine the LCM

The LCM is obtained by taking the highest power of each factor present in both polynomials. Thus, we have:
\[ \text{LCM} = (x^4 - y^4)(x^4 + y^4)(x + y) \]

### Final Answer

The LCM of the polynomials \(x^8 - y^8\) and \((x^4 - y^4)(x + y)\) is:
\[ (x^4 - y^4)(x^4 + y^4)(x + y) \]

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