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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)

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Expansion of (a ± b)3^{3}