The a–b whole cube formula (algebraic identity) is:

(a−b)3=a3−3a2b+3ab2−b3(a-b)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3}(a−b)3=a3−3a2b+3ab2−b3

It can also be written in a slightly factored form as:

(a−b)3=a3−3ab(a−b)−b3(a-b)^{3}=a^{3}-3ab(a-b)-b^{3}(a−b)3=a3−3ab(a−b)−b3

How to remember it

  • First and last terms are just the cubes: a3a^{3}a3 and −b3-b^{3}−b3.
  • Middle terms use the pattern 3, then powers of a go down (2 then 1), powers of b go up (1 then 2): −3a2b+3ab2-3a^{2}b+3ab^{2}−3a2b+3ab2.

A quick example:

(x−2y)3=x3−3x2(2y)+3x(2y)2−(2y)3=x3−6x2y+12xy2−8y3(x-2y)^{3}=x^{3}-3x^{2}(2y)+3x(2y)^{2}-(2y)^{3}=x^{3}-6x^{2}y+12xy^{2}-8y^{3}(x−2y)3=x3−3x2(2y)+3x(2y)2−(2y)3=x3−6x2y+12xy2−8y3

SEO-style quick notes

  • Focus keyword: a-b whole cube formula
  • Use it to expand or factor expressions involving the cube of a difference.

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