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} = (a^{3} – 3 x a^{2} x b + 3 x a x b^{2} – b^{3})

**Derivation of a minus b whole cube formula:**

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

= (a^{2} – 2 x a x b + b^{2}) x (a – b)

= ( a^{3} – a^{2} x b – 2 x a^{2} x b + 2 x a x b^{2} + a x b^{2} – b^{3} )

= (a^{3} – 3 x a^{2 }x b + 3 x a x b^{2} – b^{3}) (adding like terms)

Therefore, **(a – b) ^{3} = (a^{3} – 3 x a^{2 }x b + 3 x a x b^{2} – b^{3})**

On simplification, we can also write the formula as **(a – b) ^{3} = a^{3} – b^{3} – 3 x a x b x (a-b)**

**Examples:**

Solve: (2x-4)^{3}

Solution:

We know that, (a – b)^{3} = (a^{3} – 3 x a^{2 }x b + 3 x a x b^{2} – b^{3})

(2x-4)^{3} = (2x)^{3} – 3 (2x)^{2} 4 + 3 (2x) (4^{2}) – 4^{3}

(2x-4)^{3} = 8x^{3} – 48x^{2} + 96x – 64

Solve: (x – 9)^{3}

Solution:

We know that, (a – b)^{3} = (a^{3} – 3 x a^{2 }x b + 3 x a x b^{2} – b^{3})

(x-6)^{3} = (x)^{3} – 3 (x)^{2} 6 + 3 (x) (6^{2}) – 6^{3}

(x-6)^{3} = x^{3} – 18x^{2} + 108x – 216

Solve: (5y – 3x)^{3}

Solution:

We know that, (a – b)^{3} = (a^{3} – 3 x a^{2 }x b + 3 x a x b^{2} – b^{3})

(5y-3x)^{3} = (5y)^{3} – 3 (5y)^{2} 3x + 3 (5y) (3x)^{2} – (3x)^{3}

= 125y^{3} – 225xy^{2} + 135x^{2}y – 27x^{3}

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