Verify:
(i) x3 + y3 = (x + y) (x2 + y2 – xy)
(ii) x3 – y3 = (x – y) (x2 + y2 + xy)
Concept Used:
(x + y)3 = x3 + y3 + 3xy (x + y)
(x – y)3 = x3 – y3 – 3xy (x – y)
Explanation:
(i) We know that,
(x + y)3 = x3 + y3 + 3xy (x + y)
⇒ x3 + y3 = (x + y)3 – 3xy (x + y)
[Taking (x + y) common]
⇒ x3 + y3 = (x + y) [(x + y)2 – 3xy)
⇒ x3 + y3 = (x + y) [(x2 + y2 + 2xy) – 3xy]
[(x + y)2 = x2 + y2 + 2 x y]
⇒ 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)
[Taking (x – y) common]
⇒ x3 – y3 = (x – y) [(x – y)2 + 3xy)
⇒ x3 – y3 = (x – y) [(x2 + y2 – 2xy) + 3xy]
[(x – y)2 = x2 + y2 – 2 x y]
= x3 – y3 = (x – y) (x2 + y2 + xy)
Couldn't generate an explanation.
Generated by AI. May contain inaccuracies — always verify with your textbook.
