Find the LCM of `x^(8007)-1,x^(8007)+1` ?

Options

A. `(x^(16013)-1)/(2)`
B. `(x^(16014)-1)/(2)`
C. `(x^(14016)-1)/(2)`
D. `(x^(16012)-1)/(2)`

Answer

To find the LCM of \( x^{8007} - 1 \) and \( x^{8007} + 1 \), we can follow these steps:

### Step 1: Identify the two expressions
We have two expressions:
- \( a = x^{8007} - 1 \)
- \( b = x^{8007} + 1 \)

### Step 2: Use the LCM formula
The formula for the LCM of two numbers \( a \) and \( b \) is given by:
\[ \text{LCM}(a, b) = \frac{a \cdot b}{\text{HCF}(a, b)} \]

### Step 3: Calculate the product \( a \cdot b \)
Calculating the product:
\[ a \cdot b = (x^{8007} - 1)(x^{8007} + 1) = x^{8007^2} - 1^2 = x^{16014} - 1 \]

### Step 4: Find the HCF of \( a \) and \( b \)
To find the HCF of \( a \) and \( b \):
- Since \( a \) is odd and \( b \) is even, we can use the property that:
\[ \text{HCF}(x^{8007} - 1, x^{8007} + 1) = \text{HCF}(x^{8007} - 1, 2) \]
Thus, the HCF is \( 1 \) (since \( x^{8007} - 1 \) is odd).

### Step 5: Substitute into the LCM formula
Now substituting back into the LCM formula:
\[ \text{LCM}(x^{8007} - 1, x^{8007} + 1) = \frac{x^{16014} - 1}{1} = x^{16014} - 1 \]

### Step 6: Adjust for the factor of 2
Since \( x^{8007} + 1 \) is even, we divide the product by 2:
\[ \text{LCM}(x^{8007} - 1, x^{8007} + 1) = \frac{x^{16014} - 1}{2} \]

### Final Answer
Thus, the LCM of \( x^{8007} - 1 \) and \( x^{8007} + 1 \) is:
\[ \text{LCM}(x^{8007} - 1, x^{8007} + 1) = \frac{x^{16014} - 1}{2} \]

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