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

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