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

Answer

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

### Step 1: Rewrite the expression
We start with the expression:
\[ x^2 + \frac{1}{x^2} - 3 \]
We can rewrite \(-3\) as \(-2 - 1\) to help with the factorisation:
\[ x^2 + \frac{1}{x^2} - 2 - 1 \]

### Step 2: Recognize the structure
Notice that \( x^2 + \frac{1}{x^2} - 2 \) can be expressed in a different form. We can express \( x^2 + \frac{1}{x^2} \) as:
\[ \left(x - \frac{1}{x}\right)^2 \]
This is because:
\[ \left(x - \frac{1}{x}\right)^2 = x^2 - 2\cdot x\cdot\frac{1}{x} + \frac{1}{x^2} = x^2 - 2 + \frac{1}{x^2} \]

### Step 3: Substitute back
Now we can substitute back into our expression:
\[ \left(x - \frac{1}{x}\right)^2 - 1 \]

### Step 4: Apply the difference of squares
We can now use the difference of squares formula \( a^2 - b^2 = (a - b)(a + b) \):
Let \( a = \left(x - \frac{1}{x}\right) \) and \( b = 1 \):
\[ \left(x - \frac{1}{x} - 1\right)\left(x - \frac{1}{x} + 1\right) \]

### Step 5: Simplify the factors
Now we simplify each factor:
1. \( x - \frac{1}{x} - 1 = x - 1 - \frac{1}{x} \)
2. \( x - \frac{1}{x} + 1 = x + 1 - \frac{1}{x} \)

Thus, the factorised form is:
\[ \left(x - 1 - \frac{1}{x}\right)\left(x + 1 - \frac{1}{x}\right) \]

### Final Answer
The final factorised form of the expression \( x^2 + \frac{1}{x^2} - 3 \) is:
\[ \left(x - 1 - \frac{1}{x}\right)\left(x + 1 - \frac{1}{x}\right) \]