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Prove that \(x^{n}\) + \(y^{n}\) is divisible by x+y if n is odd

Can anybody prove it.Please.

Note by Ayush Choubey
2 years, 10 months ago

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Hi,@Ayush Choubey If n is odd, then \[x^{n}+y^{n}=(x+y)(x^{n-1}-x^{n-2}y+x^{n-3}y^{2}-....+y^{n-1}) \] which can be proved easily by expanding or using induction. Abhijeet Verma · 2 years ago

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Let's denote \(p(x) = x^{n}+y^{n}\). Due to remainder theorem (little Bezout's theorem) remainder will be \(p(-y)= (-y) ^{n} + y^{n}=0\), because n is odd. Therefore the remainder is 0, so p(x) is divisible by x+y. Karan Pedja · 2 years, 10 months ago

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