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Home >> Standard Identities & their applications >> (a - b)3 = a3 - b3 - 3ab(a - b) >>

## (A - B) Cube

 (a + b)2 = a2 + b2 + 2ab (a - b)2 = a2 + b2 - 2ab a2 - b2 = (a + b) (a - b) (x + a) (x + b) = x2 + x(a + b) + ab (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca (a + b)3 = a3 + b3 + 3ab(a + b) (a - b)3 = a3 - b3 - 3ab(a - b) a3 + b3 + c3 - 3abc = (a + b + c)(a2 + b2 + c2 - ab - bc - ca)

Before you understand (a - b)3 = a3 - b3 - 3ab(a - b), you are advised to read:

How to multiply Variables ?
How to multiply Constant and Variable ?
Multiplication of Polynomials ?
What is Exponential Form and Laws of Exponents ?

How identity of (a - b)3 = a3 - b3 - 3ab(a - b) is obtained

Taking LHS of the identity:
(a - b)3

This can also be written as:
= (a - b) (a - b) (a - b)

Multiply first two binomials as shown below:
= { a(a - b) - b(a - b) } (a - b)

= { a2 - ab - ab + b2 } (a - b)

Rearrange the terms in curly braces and we get:
= { a2 + b2 - ab - ab } (a - b)

Add above like terms, highlighted in red and we get:
= { a2 + b2 - 2ab } (a - b)

Multiply trinomial with binomial as shown below:
= a2(a - b) + b2(a - b) - 2ab(a - b)
= a3 - a2b + ab2 - b3 - 2a2b + 2ab2

Rearrange the terms and we get:
= a3 - b3 - a2b - 2a2b + ab2 + 2ab2

Add like terms, highlighted in green & red and we get:
= a3 - b3 - 3a2b + 3ab2

Or we can further solve it:
Take 3ab common from the above blue highlighted terms and we get:
= a3 - b3 - 3ab(a - b)

Hence, in this way we obtain the identity i.e. (a - b)3 = a3 - b3 - 3ab(a - b) = a3 - b3 - 3a2b + 3ab2

Let's try some example of this identity

Example 1: Solve (2a - 3b)3
Solution: This proceeds as:
Given polynomial (2a - 3b)3 represents identity i.e. (a - b)3
Where a = 2a and b = 3b

Now apply values of a and b on the identity i.e. (a - b)3 = a3 - b3 - 3ab(a - b) and we get:
(2a - 3b)3 = (2a)3 - (3b)3 - 3(2a) (3b)(2a - 3b)

Expand the exponential forms and we get:
= 8a3 - 27b3 - 3(2a)(3b)(2a - 3b)

Solve multiplication process and we get:
= 8a3 - 27b3 - 18ab(2a - 3b)

Hence, (2a - 3b)3 = 8a3 - 27b3 - 18ab(2a - 3b)

Example 2: Solve (5x - 6y)3
Solution: This proceeds as:
Given polynomial (5x - 6y)3 represents identity i.e. (a - b)3
Where a = 5x and b = 6y

Now apply values of a and b on the identity i.e. (a - b)3 = a3 - b3 - 3a2b + 3ab2 and we get:
(5x - 6y)3 = (5x)3 - (6y)3 - 3(5x)26y + 3(5x)(6y)2

Expand the exponential forms and we get:
= 125x3 - 216y3 - 3(25x2)(6y) + 3(5x)(36y2)

Solve multiplication process and we get:
= 125x3 - 216y3 - 450x2y + 540xy2

Hence, (5x - 6y)3 = 125x3 - 216y3 - 450x2y + 540xy2