Factorise :
` a- 3b +a^(3) - 27b^(3)`

Answer

To factorise the expression \( a - 3b + a^3 - 27b^3 \), we can follow these steps:

### Step 1: Rearrange the expression
We start by rearranging the expression to group the terms related to cubes together:
\[ a + a^3 - 3b - 27b^3 \]

### Step 2: Recognize the difference of cubes
Notice that \( a^3 - 27b^3 \) can be recognized as a difference of cubes. We can rewrite \( 27b^3 \) as \( (3b)^3 \):
\[ a^3 - (3b)^3 \]

### Step 3: Apply the difference of cubes formula
The difference of cubes can be factored using the identity:
\[ x^3 - y^3 = (x - y)(x^2 + xy + y^2) \]
Here, let \( x = a \) and \( y = 3b \). Thus, we can apply the formula:
\[ a^3 - (3b)^3 = (a - 3b)(a^2 + a(3b) + (3b)^2) \]

### Step 4: Simplify the second factor
Now, simplify the second factor:
\[ a^2 + 3ab + 9b^2 \]

### Step 5: Combine the factored terms
Now, we can combine the factored terms:
\[ a - 3b + (a - 3b)(a^2 + 3ab + 9b^2) \]
This can be rewritten as:
\[ (a - 3b)(1 + a^2 + 3ab + 9b^2) \]

### Final Result
Thus, the fully factored form of the expression \( a - 3b + a^3 - 27b^3 \) is:
\[ (a - 3b)(1 + a^2 + 3ab + 9b^2) \]