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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