Expansion of (a ± b)$^{3}$

We will discuss here about the expansion of (a ± b)$^{3}$.

(a + b)$^{3}$ = (a + b) ∙ (a + b)$^{2}$

             = (a + b)(a$^{2}$ + 2ab + b$^{2}$)

             = a(a$^{2}$ + 2ab + b$^{2}$) + b(a$^{2}$ + 2ab + b$^{2}$)

             = a$^{3}$ + 2a$^{2}$b + ab$^{2}$ + ba$^{2}$ + 2ab$^{2}$ + b$^{3}$

             = a$^{3}$ + 3a$^{2}$b + 3ab$^{2}$ + b$^{3}$.

(a - b)$^{3}$ = (a - b) ∙ (a - b)$^{2}$

            = (a - b)(a$^{2}$ - 2ab + b$^{2}$)

            = a(a$^{2}$ - 2ab + b$^{2}$) - b(a$^{2}$ - 2ab + b$^{2}$)

            = a$^{3}$ - 2a$^{2}$b + ab$^{2}$ - ba$^{2}$ + 2ab$^{2}$ - b$^{3}$

            = a$^{3}$ - 3a$^{2}$b + 3ab$^{2}$ - b$^{3}$.

Corollaries:

(a + b)$^{3}$ = a$^{3}$ + 3ab(a + b) + b$^{3}$ = a$^{3}$ + b$^{3}$ + 3ab(a + b)

(a - b)$^{3}$ = a$^{3}$ – 3ab(a - b) - b$^{3}$ = a$^{3}$ - b$^{3}$ - 3ab(a - b)

(a + b)$^{3}$ – (a$^{3}$ + b$^{3}$) = 3ab(a + b)

(a - b)$^{3}$ – (a$^{3}$ - b$^{3}$) = 3ab(a - b)

a$^{3}$ + b$^{3}$ = (a + b)$^{3}$ - 3ab(a + b)

a$^{3}$ - b$^{3}$ = (a - b)$^{3}$ + 3ab(a - b)

9th Grade Math

From Expansion of (a ± b)$^{3}$ to HOME PAGE