Verify : (i) x3 + y3 = (x + y) (x2 - xy + y2) (ii) x3 - y3 = (x - y) (x2 + xy + y2)
Verify : (i) x3 + y3 = (x + y) (x2 - xy + y2) (ii) x3 - y3 = (x - y) (x2 + xy + y2)
Answer
(i) x3 + y3 = (x + y) (x2 - xy + y2)
We know that,
(x + y)3 = x3 + y3 + 3xy(x + y)
⇒ x3 + y3 = (x + y)3 - 3xy(x + y)
⇒ x3 + y3 = (x + y)[(x + y)2 - 3xy] {Taking (x+y) common}
⇒ x3 + y3 = (x + y)[(x2 + y2 + 2xy) - 3xy]
⇒ x3 + y3 = (x + y)(x2 + y2 - xy)
(ii) x3 - y3 = (x - y) (x2 + xy + y2 )
We know that,
(x - y)3 = x3 - y3 - 3xy(x - y)
⇒ x3 - y3 = (x - y)3 + 3xy(x - y)
⇒ x3 + y3 = (x - y)[(x - y)2 + 3xy] {Taking (x-y) common}
⇒ x3 + y3 = (x - y)[(x2 + y2 - 2xy) + 3xy]
⇒ x3 + y3 = (x + y)(x2 + y2 + xy)
