Q.9. Verify:

(i) x3 + y3 = (x + y) (x2 - xy + y2)

(ii) x3 - y3 = (x - y) (x2 + xy + y2)

Solution:

(i) We know that,

(x + y)3 = x3 + y3 + 3xy(x + y)

⇒ x3 + y3 = (x + y)- 3xy(x + y)

= (x + y)[(x + y)2 - 3xy]

= (x + y)[(x2 + y+ 2xy) - 3xy]

Thus, x3 + y3 = (x + y)(x2 + y2 - xy)

(ii) We know 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]

Thus, x3 + y3 = (x + y)(x2 + y2 + xy)