Factorise :
`1029 - 3x^(3)`

Answer

To factorise the expression \( 1029 - 3x^3 \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression as it is:
\[ 1029 - 3x^3 \]

### Step 2: Factor out the common factor
We notice that \( 1029 \) can be factored as \( 3 \times 343 \) (since \( 343 = 7^3 \)). Thus, we can rewrite the expression:
\[ 1029 - 3x^3 = 3 \times 343 - 3x^3 \]
Factoring out \( 3 \):
\[ = 3(343 - x^3) \]

### Step 3: Recognize the difference of cubes
Now, we recognize that \( 343 \) is \( 7^3 \), so we can rewrite the expression inside the parentheses:
\[ 343 - x^3 = 7^3 - x^3 \]
This is a difference of cubes, which can be factored using the identity:
\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]
where \( a = 7 \) and \( b = x \).

### Step 4: Apply the difference of cubes formula
Applying the formula:
\[ 7^3 - x^3 = (7 - x)(7^2 + 7x + x^2) \]
Calculating \( 7^2 \):
\[ 7^2 = 49 \]
Thus, we have:
\[ 7^3 - x^3 = (7 - x)(49 + 7x + x^2) \]

### Step 5: Combine everything
Now, substituting back into our factored expression:
\[ 3(343 - x^3) = 3((7 - x)(49 + 7x + x^2)) \]
This simplifies to:
\[ = 3(7 - x)(49 + 7x + x^2) \]

### Final Answer
The final factorised form of the expression \( 1029 - 3x^3 \) is:
\[ 3(7 - x)(49 + 7x + x^2) \]