The factors of `x^(2)+(1)/(4x^(2))+1-2x-(1)/(x)` are

Options

A. `(x+(1)/(x))(x+(1)/(2x)-2)`
B. `(x+(1)/(2x))(x+(1)/(2x)-2)`
C. `(x+(1)/(2x))(x+(1)/(x)-2)`
D. None of these

Answer

To factor the expression \( x^2 + \frac{1}{4x^2} + 1 - 2x - \frac{1}{x} \), we can follow these steps:

### Step 1: Rewrite the expression
Start with the given expression:
\[ x^2 + \frac{1}{4x^2} + 1 - 2x - \frac{1}{x} \]

### Step 2: Identify a perfect square
Notice that \( x^2 + \frac{1}{4x^2} + 1 \) can be rewritten as a perfect square. Specifically, we can express it as:
\[ \left(x + \frac{1}{2x}\right)^2 \]
This is because:
\[ \left(x + \frac{1}{2x}\right)^2 = x^2 + 2\left(x\cdot\frac{1}{2x}\right) + \left(\frac{1}{2x}\right)^2 = x^2 + 1 + \frac{1}{4x^2} \]

### Step 3: Substitute the perfect square back
Now, substitute this back into the expression:
\[ \left(x + \frac{1}{2x}\right)^2 - 2x - \frac{1}{x} \]

### Step 4: Simplify the remaining terms
Next, we need to handle the remaining terms:
\[ -2x - \frac{1}{x} \]
We can factor out \(-2\) from these terms:
\[ -2\left(x + \frac{1}{2x}\right) \]

### Step 5: Combine the factors
Now, we can combine the two parts:
\[ \left(x + \frac{1}{2x}\right)^2 - 2\left(x + \frac{1}{2x}\right) \]
This can be factored as:
\[ \left(x + \frac{1}{2x}\right)\left(x + \frac{1}{2x} - 2\right) \]

### Step 6: Final factorization
Thus, the final factorization of the expression is:
\[ \left(x + \frac{1}{2x}\right)\left(x + \frac{1}{2x} - 2\right) \]

### Step 7: Simplifying further if needed
If we want to express it in a more standard form, we can simplify the second factor:
\[ x + \frac{1}{2x} - 2 = x - 2 + \frac{1}{2x} \]
So the final factors are:
\[ \left(x + \frac{1}{2x}\right)\left(x - 2 + \frac{1}{2x}\right) \]