Which of the following is divisible by x^2-y^ | vedclass

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(A) We know that $x^2-y^2 = (x-y)(x+y)$.For an expression to be divisible by $(x-y)(x+y)$,it must be divisible by both $(x-y)$ and $(x+y)$.Consider th

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$x^n-y^n$ for all $n \in N$

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(A) We know that $x^2-y^2 = (x-y)(x+y)$.For an expression to be divisible by $(x-y)(x+y)$,it must be divisible by both $(x-y)$ and $(x+y)$.Consider the expression $(x^n-y^n)(x^n+y^n) = x^{2n}-y^{2n}$.For $n=1$,this is $x^2-y^2$,which is divisible by $x^2-y^2$.However,looking at option $C$,we have $(x^n-y^n)(x^{2n+1}+y^{2n+1})$.If $n$ is odd,$(x^n-y^n)$ is divisible by $(x-y)$ and $(x^{2n+1}+y^{2n+1})$ is divisible by $(x+y)$.If $n$ is even,$(x^n-y^n)$ is divisible by both $(x-y)$ and $(x+y)$.Thus,the expression $(x^n-y^n)(x^n+y^n)$ or similar forms are often tested. Given the options,$(x^n-y^n)(x^n+y^n)$ is not explicitly listed,but $(x^n-y^n)(x^n+y^n) = x^{2n}-y^{2n}$ is always divisible by $x^2-y^2$ for any $n \in N$.

Which of the following is divisible by $x^2-y^2$ for all $x \neq y$?

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