Factorise : `8x^(3)+y^(3)+27z^(3)-18xyz` .

Answer

To factorize the expression \(8x^3 + y^3 + 27z^3 - 18xyz\), we can use the formula for the sum of cubes and the specific case of three variables. Here’s a step-by-step solution:

### Step 1: Identify the terms
The expression is \(8x^3 + y^3 + 27z^3 - 18xyz\). We can rewrite \(8x^3\) as \((2x)^3\) and \(27z^3\) as \((3z)^3\). Thus, we can express the equation as:
\[ (2x)^3 + y^3 + (3z)^3 - 18xyz \]

### Step 2: Recognize the form
This expression resembles the formula for the sum of cubes:
\[ a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - ac - bc) \]
where \(a = 2x\), \(b = y\), and \(c = 3z\).

### Step 3: Apply the formula
Using the values of \(a\), \(b\), and \(c\):
- \(a + b + c = 2x + y + 3z\)
- \(a^2 = (2x)^2 = 4x^2\)
- \(b^2 = y^2\)
- \(c^2 = (3z)^2 = 9z^2\)
- \(ab = (2x)(y) = 2xy\)
- \(ac = (2x)(3z) = 6xz\)
- \(bc = (y)(3z) = 3yz\)

Now, substituting these into the second part of the formula:
\[ a^2 + b^2 + c^2 - ab - ac - bc = 4x^2 + y^2 + 9z^2 - 2xy - 6xz - 3yz \]

### Step 4: Combine the results
Putting it all together, we have:
\[ 8x^3 + y^3 + 27z^3 - 18xyz = (2x + y + 3z)(4x^2 + y^2 + 9z^2 - 2xy - 6xz - 3yz) \]

### Final Factorized Form
Thus, the factorized form of the expression is:
\[ (2x + y + 3z)(4x^2 + y^2 + 9z^2 - 2xy - 6xz - 3yz) \]

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