Question

Verify the following
(i) $ {x^3} + {y^3} = (x + y)({x^2} - xy + {y^2}) $
(ii) $ {x^3} - {y^3} = (x - y)({x^2} + xy + {y^2}) $

Answer 1

Hint: Use the expansion formula of $ {(a + b)^3} $ then rearrange the terms in such a way that you get the desired result to verify the equations in the question rearranging the terms of the expression is the most important part of the solution.

Complete step-by-step answer:
(i) $ {x^3} + {y^3} = (x + y)({x^2} - xy + {y^2}) $
We know that
  $ {(x + y)^3} = {x^3} + {y^3} + 3xy(x + y) $
By rearranging it, we can write
 $ {x^3} + {y^3} = {(x + y)^3} - 3xy(x + y) $
By taking $ x + y $ common, we get
 $ {x^3} + {y^3} = [x + y][{(x + y)^2} - 3xy] $
Use the expansion formula $ {\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2} $ to get
 $ {x^3} + {y^3} = \left[ {x + y} \right]\left[ {{x^2} + 2xy + {y^2} - 3xy} \right] $
By simplifying it, we get
 $ {x^3} + {y^3} = [x + y]\left[ {{x^2} + {y^2} - xy} \right] $
By rearranging it, we get
 $ {x^3} + {y^3} = [x + y]\left[ {{x^2} - xy + {y^2}} \right] $
 $ = R.H.S $
Hence $ {x^3} + {y^3} = (x + y)({x^2} - xy + {y^2}) $ is verified
(ii) Now, we have to verify, $ {x^3} - {y^3} = (x - y)({x^2} + xy + {y^2}) $
We know that
 $ {(x - y)^3} = {x^3} - {y^3} - 3xy(x - y) $
By rearranging it, we get
 $ {x^3} - {y^3} = {(x - y)^3} + 3xy(x - y) $
By taking $ (x - y) $ common, we get
 $ {x^3} - {y^3} = (x - y)\left[ {{{\left( {x - y} \right)}^2} + 3xy} \right] $
By expanding it using $ {\left( {a - b} \right)^2} = {a^2} - 2ab + {b^2} $ , we get
 $ {x^3} - {y^3} = \left[ {x - y} \right]\left[ {{x^2} - 2xy + {y^2} + 3xy} \right] $
On simplifying it, we get
 $ {x^3} - {y^3} = \left[ {x - y} \right]\left[ {{x^2} + xy + {y^2}} \right] $
 $ = R.H.S $
Hence it is verified that $ {x^3} - {y^3} = (x - y)({x^2} + xy + {y^2}) $

Note: All we did in this question is to manipulate the terms of formula to get the desired result. You need to understand how to manipulate any standard formula so solve such kinds of questions. But you cannot solve such questions if you don’t know the formula in the first place.