Factorise `x^(2)+(1)/(x^(2))-3`.

Options

A. `(x-(1)/(x)+a)(x+(1)/(x)-1)`
B. `(x+(1)/(x)+a)(x-(1)/(x)-1)`
C. `(x-(1)/(x)+a)(x-(1)/(x)+1)`
D. `(x-(1)/(x)+a)(x-(1)/(x)-1)`

Explanation

To factorise the expression \( x^2 + \frac{1}{x^2} - 3 \), we can follow these steps:

### Step 1: Rewrite the expression
Start with the expression:
\[ x^2 + \frac{1}{x^2} - 3 \]

### Step 2: Combine \( x^2 \) and \( \frac{1}{x^2} \)
We can rewrite \( x^2 + \frac{1}{x^2} \) using the identity:
\[ x^2 + \frac{1}{x^2} = \left( x + \frac{1}{x} \right)^2 - 2 \]
So, we can substitute this into our expression:
\[ \left( x + \frac{1}{x} \right)^2 - 2 - 3 = \left( x + \frac{1}{x} \right)^2 - 5 \]

### Step 3: Recognize the difference of squares
Now, we have:
\[ \left( x + \frac{1}{x} \right)^2 - 5 \]
This can be factored using the difference of squares formula \( a^2 - b^2 = (a - b)(a + b) \):
\[ \left( x + \frac{1}{x} - \sqrt{5} \right)\left( x + \frac{1}{x} + \sqrt{5} \right) \]

### Step 4: Write the final factorized form
Thus, the factorized form of the expression \( x^2 + \frac{1}{x^2} - 3 \) is:
\[ \left( x + \frac{1}{x} - \sqrt{5} \right)\left( x + \frac{1}{x} + \sqrt{5} \right) \]