a-b whole cube

June 28, 2026

The standard identity for a − b whole cube 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.

(a − b)³ formula in words

It is the cube of the difference of two terms a and b.

Expanded form: (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.

Alternate compact form: (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.

Quick derivation idea

Start with multiplying (a−b)(a-b)(a−b) three times:
(a−b)3=(a−b)(a−b)(a−b)(a-b)^{3}=(a-b)(a-b)(a-b)(a−b)3=(a−b)(a−b)(a−b).

First, use the square identity: (a−b)2=a2−2ab+b2(a-b)^{2}=a^{2}-2ab+b^{2}(a−b)2=a2−2ab+b2.

Then multiply this by (a−b)(a-b)(a−b):
(a2−2ab+b2)(a−b)(a^{2}-2ab+b^{2})(a-b)(a2−2ab+b2)(a−b).

On expansion and combining like terms, you get:
a3−3a2b+3ab2−b3a^{3}-3a^{2}b+3ab^{2}-b^{3}a3−3a2b+3ab2−b3.

How to remember it

A simple way many students use now (and in current online classes and videos) is to compare it with (a+b)3(a+b)^{3}(a+b)3 and just flip the alternating signs.

For (a+b)3(a+b)^{3}(a+b)3: a3+3a2b+3ab2+b3a^{3}+3a^{2}b+3ab^{2}+b^{3}a3+3a2b+3ab2+b3.

For (a−b)3(a-b)^{3}(a−b)3: keep the same coefficients but alternate signs:
a3−3a2b+3ab2−b3a^{3}-3a^{2}b+3ab^{2}-b^{3}a3−3a2b+3ab2−b3.

Tiny example

Let a = 2, b = 1.

Direct cube: (2−1)3=13=1(2-1)^{3}=1^{3}=1(2−1)3=13=1.
Using the identity:
23−3⋅22⋅1+3⋅2⋅12−13=8−12+6−1=12^{3}-3\cdot 2^{2}\cdot 1+3\cdot 2\cdot 1^{2}-1^{3}=8-12+6-1=123−3⋅22⋅1+3⋅2⋅12−13=8−12+6−1=1.

Both give 1, so the identity checks out.

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