Question

Factorise the following:

4x² – 9y² – 8x³ + 27y³

Answer 1

Given expression: 4x² – 9y² – 8x³ + 27y³

Step-wise calculation:

Step 1: Group terms for easier factorization:

(4x² – 9y²) – (8x³ – 27y³)

Step 2: Recognize that (4x² – 9y²) is a difference of squares:

4x² – 9y² = (2x)² – (3y)²

4x² – 9y² = (2x – 3y)(2x + 3y)

Step 3: Recognize that (8x³ – 27y³) is a difference of cubes:

8x³ – 27y³ = (2x)³ – (3y)³

8x³ – 27y³ = (2x – 3y)(4x² + 6xy + 9y²)

Step 4: Substitute these back into the original expression:

(2x – 3y)(2x + 3y) – (2x – 3y)(4x² + 6xy + 9y²)

Step 5: Factor out the common factor (2x – 3y):

(2x – 3y) [(2x + 3y) – (4x² + 6xy + 9y²)]

Step 6: Simplify inside the brackets:

(2x + 3y) – (4x² + 6xy + 9y²) = 2x + 3y – 4x² – 6xy – 9y²

Step 7: Write the final factorised form:

(2x – 3y) (2x + 3y – 4x² – 6xy – 9y²)