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# Prove that $$x^{n}$$ + $$y^{n}$$ is divisible by x+y if n is odd

Note by Ayush Choubey
3 years, 3 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.

- 2 years, 6 months ago

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.

- 3 years, 3 months ago