Factorise: `x^(4) + x^(2)+1`

Answer

To factorise the expression \( x^4 + x^2 + 1 \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ x^4 + x^2 + 1 \]
We can add and subtract \( x^2 \) to help us rearrange the terms:
\[ x^4 + x^2 + 1 = x^4 + 2x^2 + 1 - x^2 \]

### Step 2: Group the terms
Now we can group the first three terms:
\[ = (x^4 + 2x^2 + 1) - x^2 \]

### Step 3: Recognize a perfect square
The expression \( x^4 + 2x^2 + 1 \) can be recognized as a perfect square:
\[ = (x^2 + 1)^2 - x^2 \]

### Step 4: Apply the difference of squares
Now we have a difference of squares, which can be factored using the identity \( a^2 - b^2 = (a + b)(a - b) \):
Let \( a = x^2 + 1 \) and \( b = x \):
\[ = (x^2 + 1 + x)(x^2 + 1 - x) \]

### Step 5: Write the final factorized form
Thus, the factorized form of \( x^4 + x^2 + 1 \) is:
\[ (x^2 + x + 1)(x^2 - x + 1) \]

### Summary of the Factorization
The final factorization of the expression \( x^4 + x^2 + 1 \) is:
\[ (x^2 + x + 1)(x^2 - x + 1) \]

---