Math Doubts

$(a-b)^3$ identity

Formula

$(a-b)^3$ $\,=\,$ $a^3-b^3-3ab(a-b)$

Introduction

Let $a$ and $b$ be two variables in algebraic form and they represent two terms mathematically. The difference between them is expressed in mathematical form as $a-b$, which is basically an algebraic expression and also a binomial. The cube of difference of two terms or a binomial is written in the below form in mathematics.

$(a-b)^3$

The $a$ minus $b$ whole cube is equal to $a$ cubed minus $b$ cubed minus $3ab$ times $a$ minus $b$.

$(a-b)^3$ $\,=\,$ $a^3-b^3-3ab(a-b)$

This algebraic identity can be written in the following form too.

$\implies$ $(a-b)^3$ $\,=\,$ $a^3-b^3-3a^2b+3ab^2$

Generally, the $a$ minus $b$ whole cubed algebraic identity is called by the following three ways in mathematics.

  1. The cube of difference between two terms identity or simply the cube of difference identity.
  2. The cube of a binomial formula.
  3. The special binomial product rule.

Usage

In fact, the cube of difference algebraic identity is used in two special cases in mathematics.

Expansion

The cube of difference between any two terms is expanded as the subtraction of three times the product of both terms and the subtraction of the second term from the first term, from the subtraction of the cube of the second term from the cube of the first term.

$\implies$ $(a-b)^3$ $\,=\,$ $a^3-b^3-3ab(a-b)$

Simplification

The subtraction of three times the product of both terms and the subtraction of the second term from the first term, from the subtraction of the cube of the second term from the cube of the first term is simplified as the cube of difference between any two terms.

$\implies$ $a^3-b^3-3ab(a-b)$ $\,=\,$ $(a-b)^3$

Proofs

The $a$ minus $b$ whole cube identity can be proved mathematically in two different mathematical approaches.

Algebraic method

Learn how to derive the expansion of $a$ minus $b$ whole cubed identity by the product of three same difference basis binomials.

Learn more
Geometric method

Learn how to derive the expansion of $a$ minus $b$ whole cube formula geometrically by the volume of a cube.

Learn more