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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)
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06/07/2021 11:55 am
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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})
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06/07/2021 11:58 am
(i) x^{3}+y^{3 }= (x+y)(x^{2}–xy+y^{2})
We know that, (x+y)^{3} = x^{3}+y^{3}+3xy(x+y)
⇒ x^{3}+y^{3 }= (x+y)^{3}–3xy(x+y)
⇒ x^{3}+y^{3 }= (x+y)[(x+y)^{2}–3xy]
Taking (x+y) common ⇒ x^{3}+y^{3 }= (x+y)[(x^{2}+y^{2}+2xy)–3xy]
⇒ x^{3}+y^{3 }= (x+y)(x^{2}+y^{2}–xy)
(ii) x^{3}–y^{3 }= (x–y)(x^{2}+xy+y^{2})
We know that,(x–y)^{3} = x^{3}–y^{3}–3xy(x–y)
⇒ x^{3}−y^{3 }= (x–y)^{3}+3xy(x–y)
⇒ x^{3}−y^{3 }= (x–y)[(x–y)^{2}+3xy]
Taking (x+y) common
⇒ x^{3}−y^{3 }= (x–y)[(x^{2}+y^{2}–2xy)+3xy]
⇒ x^{3}+y^{3 }= (x–y)(x^{2}+y^{2}+xy)
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