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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