` ( x + (1)/( x)) (x - (1)/( x)) ( x^(2) + (1)/( x^(2)) - 1) ( x^(2) + (1)/(x^(2))+ 1)` is equal to

Options

A. ` x^(6) + (1)/(x^(6))`
B. `x^(8) + (1)/(x^(8))`
C. ` x^(8) - (1)/( x^(8))`
D. ` x^(6) - (1)/( x^(6))`

Answer

To solve the expression \( (x + \frac{1}{x})(x - \frac{1}{x})(x^2 + \frac{1}{x^2} - 1)(x^2 + \frac{1}{x^2} + 1) \), we can simplify it step by step using algebraic identities.

### Step 1: Simplify the first part using the difference of squares
We can use the identity \( a^2 - b^2 = (a + b)(a - b) \).

Let:
- \( a = x \)
- \( b = \frac{1}{x} \)

So, we have:
\[ (x + \frac{1}{x})(x - \frac{1}{x}) = x^2 - \left(\frac{1}{x}\right)^2 = x^2 - \frac{1}{x^2} \]

### Step 2: Rewrite the expression
Now, we can rewrite the original expression:
\[ (x^2 - \frac{1}{x^2})(x^2 + \frac{1}{x^2} - 1)(x^2 + \frac{1}{x^2} + 1) \]

### Step 3: Simplify the second part using the sum and difference of squares
Now, we can simplify the product \( (x^2 + \frac{1}{x^2} - 1)(x^2 + \frac{1}{x^2} + 1) \) using the same identity.

Let:
- \( c = x^2 + \frac{1}{x^2} \)

So, we have:
\[ (c - 1)(c + 1) = c^2 - 1 \]

### Step 4: Substitute back
Now substituting back, we get:
\[ (x^2 - \frac{1}{x^2})(c^2 - 1) \]

### Step 5: Calculate \( c^2 \)
We need to find \( c^2 \):
\[ c = x^2 + \frac{1}{x^2} \]
Thus,
\[ c^2 = \left(x^2 + \frac{1}{x^2}\right)^2 = x^4 + 2 + \frac{1}{x^4} \]
So,
\[ c^2 - 1 = x^4 + 2 + \frac{1}{x^4} - 1 = x^4 + 1 + \frac{1}{x^4} \]

### Step 6: Final expression
Now we can write the final expression:
\[ (x^2 - \frac{1}{x^2})(x^4 + 1 + \frac{1}{x^4}) \]

### Step 7: Expand the final expression
Now we expand:
\[ x^2(x^4 + 1 + \frac{1}{x^4}) - \frac{1}{x^2}(x^4 + 1 + \frac{1}{x^4}) = x^6 + x^2 + 1 - \left(\frac{x^2}{x^4} + \frac{1}{x^2} + \frac{1}{x^6}\right) \]
This simplifies to:
\[ x^6 - \frac{1}{x^6} + x^2 - \frac{1}{x^2} \]

### Final Result
Thus, the final result of the expression is:
\[ x^6 - \frac{1}{x^6} \]

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