Verify : (i) x 3 + y 3 = (x + y) (x 2 – xy + y 2 ) (ii) x 3 – y 3 = (x – y) (x 2 + xy + y 2 ).
Solution and Explanation
(i) It is known that,
(x + y)3 = x3 + y3 + 3xy(x + y)
x3 + y3 = (x + y)3 - 3xy (x + y)
= (x + y) [(x + y)2 - 3xy]
= (x + y) (x2 + y2 + 2xy - 3xy)
= (x + y) (x2 + y2 - xy)
= (x + y) (x2 - xy + y2)
(ii) It is known that,
(x - y)3 = x3 - y3 - 3xy (x - y)
x3 - y3 = (x - y)3 + 3xy (x - y)
= (x - y) [(x - y)2 + 3xy]
= (x - y) (x2 + y2 - 2xy + 3xy)
= (x - y) (x2 + y2 + xy)
= (x - y) (x2 + xy + y2).
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