Factorisation of 8X³ + 27Y³ + 36X²Y + 54XY²
To factorise the given expression, we will first look for common factors in pairs of terms and then apply algebraic identities to simplify the expression.
1. Common Factor of X and Y:
- The given expression can be written as:
8X³ + 27Y³ + 36X²Y + 54XY²
- Factor out the common factor of X and Y:
X³(8 + 36X + 54Y) + Y³(27 + 36X)
2. Apply Algebraic Identities:
- We notice that 8, 36, and 54 form an arithmetic progression with a common difference of 28.
- Similarly, 27 and 36 form an arithmetic progression with a common difference of 9.
- We can rewrite the expression as:
X³(2³ + 3³) + Y³(3³ + 2³) + 18XY(2X + 3Y)
3. Apply Sum of Cubes Formula:
- Using the formula a³ + b³ = (a + b)(a² - ab + b²), we can factorise the expression further:
(2X + 3Y)(4X² - 6XY + 9Y²) + (3X + 2Y)(9X² - 6XY + 4Y²)
Therefore, the factorised form of 8X³ + 27Y³ + 36X²Y + 54XY² is:
(2X + 3Y)(4X² - 6XY + 9Y²) + (3X + 2Y)(9X² - 6XY + 4Y²)