Factorise: `x ^(8) - x ^(4) - 72`

Answer

To factorize the expression \( x^8 - x^4 - 72 \), we will follow these steps:

### Step 1: Substitute \( y = x^4 \)
Let’s first make the substitution \( y = x^4 \). This transforms our expression into:
\[ y^2 - y - 72 \]

### Step 2: Factor the quadratic expression
Next, we need to factor the quadratic expression \( y^2 - y - 72 \). We are looking for two numbers that multiply to \(-72\) (the constant term) and add up to \(-1\) (the coefficient of \(y\)).

The factors of \(-72\) that satisfy this condition are \(8\) and \(-9\) because:
\[ 8 \times (-9) = -72 \quad \text{and} \quad 8 + (-9) = -1 \]

### Step 3: Rewrite the quadratic using the factors
We can now rewrite the quadratic as:
\[ y^2 - 9y + 8y - 72 \]
Grouping the terms, we have:
\[ (y^2 - 9y) + (8y - 72) \]

### Step 4: Factor by grouping
Now, we factor by grouping:
\[ y(y - 9) + 8(y - 9) \]
This gives us:
\[ (y - 9)(y + 8) \]

### Step 5: Substitute back \( y = x^4 \)
Now, we substitute back \( y = x^4 \):
\[ (x^4 - 9)(x^4 + 8) \]

### Step 6: Factor further if possible
The term \( x^4 - 9 \) can be factored further since it is a difference of squares:
\[ x^4 - 9 = (x^2 - 3)(x^2 + 3) \]

### Final Factorization
Thus, the complete factorization of the original expression \( x^8 - x^4 - 72 \) is:
\[ (x^2 - 3)(x^2 + 3)(x^4 + 8) \]

### Summary of the Factorization
The final factorized form is:
\[ (x^2 - 3)(x^2 + 3)(x^4 + 8) \]

---