# a minus b whole cube formula explained

Learn and know what is the formula of a minus b whole cube in algebra chapter. Do you know this formula we can give as an example of identity in math. Get the a minus b whole cube formula by doing a small modification i.e. just replacing the “plus” sign to the “minus” sign in the a plus b whole cube formula. Otherwise in the normal method also we can derive the a minus b whole cube formula. So first we will know what is the (a-b)3 formula and later we will see the derivation of it.

## Formula for a minus b whole cube as follows:

a minus b whole cube is written as a cube minus 3 a square b plus 3 a b square minus b cube. Mathematically we can express the formula as (a – b)3 = (a3 – 3 x a2 x b + 3 x a x b2 – b3)

## Derivation of a minus b whole cube formula:

We can write (a – b)3 =  (a – b)2 x (a – b)

= (a2 – 2 x a x b + b2) x (a – b)

= ( a3 – a2 x b – 2 x a2 x b + 2 x a x b2 + a x b2 – b3 )

= (a3 – 3 x a2 x b + 3 x a x b2 – b3) (adding like terms)

Therefore, (a – b)3 = (a3 – 3 x a2 x b + 3 x a x b2 – b3)

On simplification, we can also write the formula as (a – b)3 = a3 – b3 – 3 x a x b x (a-b)

### Examples:

Solve: (2x-4)3

Solution:

We know that, (a – b)3   = (a3 – 3 x a2 x b + 3 x a x b2 – b3)

(2x-4)3   = (2x)3 – 3 (2x)2 4 + 3 (2x) (42) – 43

(2x-4)3 = 8x3 – 48x2 + 96x – 64

Solve: (x – 9)3

Solution:

We know that, (a – b)3 = (a3 – 3 x a2 x b + 3 x a x b2 – b3)

(x-6)3 = (x)3 – 3 (x)2 6 + 3 (x) (62) – 63

(x-6)3 = x3 – 18x2 + 108x – 216

Solve: (5y – 3x)3

Solution:

We know that, (a – b)3   =  (a3 – 3 x a2 x b + 3 x a x b2 – b3)

(5y-3x)3 = (5y)3 – 3 (5y)2 3x + 3 (5y) (3x)2 – (3x)3

= 125y3 – 225xy2 + 135x2y – 27x3 