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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)(a2 + 2ab + b2)

             = a(a2 + 2ab + b2) + b(a2 + 2ab + b2)

             = a3 + 2a2b + ab2 + ba2 + 2ab2 + b3

             = a3 + 3a2b + 3ab2 + b3.

(a - b)3 = (a - b) ∙ (a - b)2

            = (a - b)(a2 - 2ab + b2)

            = a(a2 - 2ab + b2) - b(a2 - 2ab + b2)

            = a3 - 2a2b + ab2 - ba2 + 2ab2 - b3

            = a3 - 3a2b + 3ab2 - b3.


Corollaries:

(a + b)3 = a3 + 3ab(a + b) + b3 = a3 + b3 + 3ab(a + b)


(a - b)3 = a3 – 3ab(a - b) - b3 = a3 - b3 - 3ab(a - b)


(a + b)3 – (a3 + b3) = 3ab(a + b)


(a - b)3 – (a3 - b3) = 3ab(a - b)


a3 + b3 = (a + b)3 - 3ab(a + b)


a3 - b3 = (a - b)3 + 3ab(a - b)







9th Grade Math

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