This document summarizes the steps to show that a3 + b3 can be rewritten as (a + b)(a2 - ab + b2). It starts with the original model of the expression using blocks of size a and b. It then rearranges the blocks into base areas of a2, -ab, and b2 with a height of a + b. Bringing these pieces together provides the final factorization.

1. a3 + b3
By
Swati Sircar
(longer version)

2. a3 + b3
a
a
a
b
b
b

3. a3 + b3
b
b
b
a
a - b
a
bb
a - b

4. a3 + b3
rearranging
pieces
(a – b)a2
(a – b)ab
ab2
b3

5. rearranging
pieces
a3 + b3
a
a – b
a

6. rearranging
pieces
a3 + b3
a
a – b
aa + b
b

7. rearranging
pieces
a3 + b3
a
a – b
a
a + b
b b

8. pieces
rearranged
a3 + b3
a
a – b
a
a + b
b b
b

9. base area

10. base area: a2
a2
a
a

11. base area: a2
a - b
a
b

12. base area:
a2
– ab
ab
a - b
a

13. base area:
a2 – ab
a2 – ab

14. base area:
a2 – ab + b2a2 – ab
b2

15. base area:
a2 – ab + b2
height: a + b
(a + b)(a2 – ab + b2)
= a3 + b3
a2 – ab
b2
a + b

16. model
(a – b)a2
(a – b)ab
ab2
b3
a3

17. Original model – VASCSC
Model modified in 2014 with few extra blocks
Model can be flattened out
Shot at APU