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Q.

What is the formula for  a3-b3?


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a

(a-b)3+3ab(a-b)

b

(a-b)3-3ab(a-b)

c

(a-b)3+3ab(a+b)

d

(a-b)3-3ab(a+b) 

answer is A.

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Detailed Solution

Formula given: a3-b3 .
Now, we can clearly see that the power of both the terms ‘a’ and ‘b’ in the expression a3-b3 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=  (a2−ab−ba+b2)(a−b)
(a-b)3 = (a2−2ab+b2)(a−b)
Now, multiplying and simplifying the remaining term in the R.H.S, we get,
(a-b)3= (a3a2b−2a2b+2a2b+b2a−b3)(a−b)
(a-b)3= =(a3−3a2b+3ab2b3)
(a-b)3= (a3-b3) +3ab2−3a2b
⇒ (a3-b3) = (a-b)3−3a2b+3ab2
⇒ (a3-b3)  = (a-b)3+3ab (a-b)
So, the above expression is the required formula for the provided expression (a3-b3).
 
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