Factorise: `a ^(6) - 7a ^(3) -18`

Answer

To factorize the expression \( a^6 - 7a^3 - 18 \), we can follow these steps:

### Step 1: Substitute
Let \( x = a^3 \). Then, the expression becomes:
\[ x^2 - 7x - 18 \]

### Step 2: Factor the Quadratic
We need to factor the quadratic expression \( x^2 - 7x - 18 \). We are looking for two numbers that multiply to \(-18\) (the constant term) and add up to \(-7\) (the coefficient of \(x\)).

### Step 3: Find the Numbers
The factors of \(-18\) that add up to \(-7\) are \(-9\) and \(2\):
- \(-9 \times 2 = -18\)
- \(-9 + 2 = -7\)

### Step 4: Rewrite the Expression
Using these factors, we can rewrite the quadratic:
\[ x^2 - 9x + 2x - 18 \]

### Step 5: Group the Terms
Now, we group the terms:
\[ (x^2 - 9x) + (2x - 18) \]

### Step 6: Factor by Grouping
Factor out the common terms from each group:
\[ x(x - 9) + 2(x - 9) \]

### Step 7: Factor Out the Common Binomial
Now, we can factor out the common binomial factor \((x - 9)\):
\[ (x - 9)(x + 2) \]

### Step 8: Substitute Back
Now, substitute back \(x = a^3\):
\[ (a^3 - 9)(a^3 + 2) \]

### Final Answer
Thus, the factorized form of the expression \( a^6 - 7a^3 - 18 \) is:
\[ (a^3 - 9)(a^3 + 2) \]