If 9x$^2$ + 6xy + 4y$^2$ is a factor of 27x$^3$ - 8y$^3$, find the other factor.

Analyze the question: We're told that 9x² + 6xy + 4y² is one factor of 27x³ - 8y³. We need the other factor.

Concept: This is the difference of two cubes:

a³ - b³ = (a - b)(a² + ab + b²)

Solution:

1. Write each term as a perfect cube: • 27x³ = (3x)³, so a = 3x • 8y³ = (2y)³, so b = 2y

2. Apply the formula:

27x³ - 8y³ = (3x - 2y)(a² + ab + b²)

3. Check that the second bracket matches the given factor: • a² = (3x)² = 9x² • ab = (3x)(2y) = 6xy • b² = (2y)² = 4y²

So a² + ab + b² = 9x² + 6xy + 4y², which matches the given factor exactly.

4. Therefore the other factor is 3x - 2y.

Verification by expanding: (3x - 2y)(9x² + 6xy + 4y²) = 27x³ + 18x²y + 12xy² - 18x²y - 12xy² - 8y³ = 27x³ - 8y³ ✓

Answer: The other factor is 3x - 2y, which is option D.

Tip: Remember the pattern "SOAP" for cube factorisations:

• Same sign as the original in the first bracket
• Opposite sign in the middle of the second bracket
• Always Positive for the last term

For a³ - b³: (a - b)(a² + ab + b²). For a³ + b³: (a + b)(a² - ab + b²).