Difference of two cubes

Non-AfC reviewer comment: this is not fundamentally a separate topic from Sum of two cubes. --JBL (talk) 20:39, 4 January 2023 (UTC) ---

Visual proof of the formulas for the sum and difference of two cubes

In mathematics, the difference of two cubes is a cubed number subtracted from another cubed number.

Factorization

Every difference of cubes may be factored according to the identity

a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})

Starting with the expression, $a^{2} + a b + b^{2}$ is multiplied by a and b.

(a - b) (a^{2} + a b + b^{2}) = a (a^{2} + a b + b^{2}) + b (a^{2} + a b + b^{2})

By distributing a and b to $a^{2} + a b + b^{2}$ , we get

a^{3} + a^{2} b + a b^{2} - b a^{2} - b^{2} a - b^{3}

By combining, both middle terms cancel:

a^{2} b - b a^{2} + a b^{2} - b^{2} a = 0

leaving $a^{3} - b^{3}$

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