Q. 213.8( 52 Votes )

Verify:

(i) x3 + y3 = (x + y) (x2 + y2 – xy)

(ii) x3 – y3 = (x – y) (x2 + y2 + xy)

Answer :

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)


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