Verify identity

a3
+ b3
= (a + b) (a2
– ab + b2
)
Please answer in LHS = RHS format

September 17, 2021

By Mackenzie

2 thoughts on “Verify identity<br /><br />a3<br /> + b3<br /> = (a + b) (a2<br /> – ab + b2<br /> )<br />Please answer in LHS = RHS format<br”

1. Since the expression is derived from (a+b)^3

   So let us expand it

   (a+b)^3

   = (a+b) (a+b) (a+b)

   ={(a+b) (a+b)} (a+b)

   ={a(a+b) + b(a+b)} (a+b)

   =(a^2 + ab + ab + b^2) (a+b)

   =(a^2 + b^2 + 2ab) (a+b)

   =a^2(a+b) + b^2(a+b) + 2ab(a+b)

   =a^3 + a^2b + ab^2 + b^3 + 2a^2b + 2ab^2

   =a^3 + b^3 + 3a^2b + 3ab^2

   =a^3 + b^3 + 3ab(a+b)

   Now when we have expanded (a+b)^3 = a^3 + b^3 + 3ab(a+b)

   We can equate it

   (a+b)^3 = a^3 + b^3 + 3ab(a+b)

   (a+b)^3 – 3ab(a+b) = a^3 + b^3

   a^3 + b^3 = (a+b)^3 – 3ab(a+b)

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2. Answer:

   You know that,

   (a + b)³ = a³ + 3ab(a + b) + b³

   then,

   a³ + b³ = (a + b)³ – 3ab(a + b)

   = (a + b)[(a + b)² – 3ab]

   = (a + b)(a² + 2ab + b² – 3ab)

   = (a + b)(a² – ab + b² )

   Reply