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Mathematics Question on limits and derivatives

Which of the following is divisible by x2−y2,∀x≠yx^{2}-y^{2}, \forall x\ne y?

A

xn−yn∀   n∈Nx^{n}-y^{n} \forall\,\,\, n\in N

B

xn+yn∀   n∈Nx^{n}+y^{n} \forall\,\,\, n\in N

C

(xn−yn) (x2n+1+y2n+1)∀   n∈N(x^{n}-y^{n}) \, (x^{2n+1}+y^{2n+1}) \forall\,\,\, n\in N

D

(xn−yn) (xm+ym)∀m,n∈N(x^{n}-y^{n}) \, (x^m+y^m) \forall m, n\in N

Answer

(xn−yn) (x2n+1+y2n+1)∀   n∈N(x^{n}-y^{n}) \, (x^{2n+1}+y^{2n+1}) \forall\,\,\, n\in N

Explanation

Solution

∵xn−yn \because x^{n}-y^{n} is divisible by (x+y)(x−y)(x+y)(x-y), if
n=n= even
Similarly, xn−ynx^{n}-y^{n} is divisible by (x−y)(x-y),
if n=n= odd and (x2n+1+y2n+1)\left(x^{2 n+1}+y^{2 n+1}\right) is divisible by (x+y)(x+y).