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What is the formula for $a^{3} - b^{3}$ ?

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{(a - b)}^{3} + 3 ab (a - b)

{(a - b)}^{3} - 3 ab (a - b)

{(a - b)}^{3} + 3 ab (a + b)

{(a - b)}^{3} - 3 ab (a + b)

answer is A.

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Detailed Solution

Formula given:

a^{3} - b^{3}

.
Now, we can clearly see that the power of both the terms ‘a’ and ‘b’ in the expression

a^{3} - b^{3}

is 3 and there is a minus sign between the terms. So, let us consider the difference (a – b). Now, multiplying (a – b) three times with itself, we have,
⇒(a−b)(a−b)(a−b) =

{(a - b)}^{3}

⇒

{(a - b)}^{3}

= (a−b)(a−b)(a−b)
Multiplying the terms in the R.H.S by taking two at a time, we get,
⇒

{(a - b)}^{3}

= (

a^{2}

−ab−ba+

b^{2}

)(a−b)
⇒

{(a - b)}^{3}

= (

a^{2}

−2ab+

b^{2}

)(a−b)
Now, multiplying and simplifying the remaining term in the R.H.S, we get,
⇒

{(a - b)}^{3}

= (

a^{3}

−

a^{2}

b−2

a^{2}

b+2

a^{2}

b^{2}

a−

b^{3}

)(a−b)
⇒

{(a - b)}^{3}

= =(

a^{3}

−3

a^{2}

b+3a

b^{2}

−

b^{3}

)
⇒

{(a - b)}^{3}

= (

a^{3} - b^{3}

) +3a

b^{2}

−3

a^{2}

b
⇒ (

a^{3} - b^{3}

) =

{(a - b)}^{3}

−3

a^{2}

b+3a

b^{2}

⇒ (

a^{3} - b^{3}

) =

{(a - b)}^{3}

+3ab (a-b)
So, the above expression is the required formula for the provided expression (

a^{3} - b^{3}

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