The factors of `[(2)/(x^(4))-(1)/(x^(2))]` will be

Options

A. `((sqrt(2))/(x^(4))+(1)/(x))((sqrt(2))/(x^(4))-(1)/(x))`
B. `((sqrt(2))/(x^(2))+(1)/(x))((sqrt(2))/(x^(2))-(1)/(x))`
C. `((sqrt(2))/(x)+(1)/(x))((sqrt(2))/(x)-(1)/(x))`
D. None of these

Answer

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

### Step 1: Rewrite the expression
We start with the expression:
\[ \frac{2}{x^4} - \frac{1}{x^2} \]
To combine these fractions, we need a common denominator. The common denominator here is \(x^4\). We can rewrite the second term:
\[ \frac{1}{x^2} = \frac{x^2}{x^4} \]
So, we rewrite the expression as:
\[ \frac{2}{x^4} - \frac{x^2}{x^4} = \frac{2 - x^2}{x^4} \]

### Step 2: Factor the numerator
Now, we focus on factoring the numerator \(2 - x^2\). We can recognize that this expression can be rearranged:
\[ 2 - x^2 = \sqrt{2}^2 - (\sqrt{x})^2 \]
This is a difference of squares, which can be factored using the identity \(a^2 - b^2 = (a + b)(a - b)\):
\[ 2 - x^2 = (\sqrt{2} + \sqrt{x})(\sqrt{2} - \sqrt{x}) \]

### Step 3: Write the complete factorization
Now, substituting back into our expression, we have:
\[ \frac{2 - x^2}{x^4} = \frac{(\sqrt{2} + \sqrt{x})(\sqrt{2} - \sqrt{x})}{x^4} \]
This can also be expressed as:
\[ \frac{(\sqrt{2} + \sqrt{x})(\sqrt{2} - \sqrt{x})}{x^2 \cdot x^2} = \frac{(\sqrt{2} + \sqrt{x})(\sqrt{2} - \sqrt{x})}{x^2} \cdot \frac{1}{x^2} \]

### Final Answer
Thus, the factors of \(\frac{2}{x^4} - \frac{1}{x^2}\) are:
\[ \frac{(\sqrt{2} + \frac{1}{\sqrt{x}})(\sqrt{2} - \frac{1}{\sqrt{x}})}{x^4} \]