Question

Factorise the following:

x³ + 8y³ + x² – 4y²

Answer 1

Given: x³ + 8y³ + x² – 4y²

Step-wise calculation:

1. Group the terms as sums of cubes and other terms:

x³ + 8y³ + x² – 4y² = (x³ + 8y³) + (x² – 4y²)

2. Recognize the sums and differences of cubes and squares:

x³ + 8y³  = x³  + (2y)³

x² – 4y² = x² – (2y)²

3. Factor the sum of cubes using:

a³ + b³ = (a + b)(a² – ab + b²) 

Apply with (a = x), (b = 2y):

x³ + 8y³ = (x + 2y)(x² – 2xy + 4y²)

4. Factor the difference of squares using:

a² – b² = (a – b)(a + b) 

Apply with (a = x), (b = 2y):

x² – 4y² = (x – 2y)(x + 2y)

5. Rewrite the expression grouping the factors:

(x + 2y)(x² – 2xy + 4y²) + (x – 2y)(x + 2y)

6. Factor out the common factor (x + 2y): 

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

7. Simplify the expressions inside the bracket:

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

8. Final factorized form:

(x + 2y)(x² – 2xy + 4y² + x – 2y)