I'm trying to prove that x^{n}+y^{n}is divisible by (x+y), when x and y are integers, and n is a positive, odd integer. I previously proved that x^{n}-y^{n }was divisible by (x-y) with induction, so I figured a similar method would make sense here, but I end up going in circles... this is what I tried:

Since n is an positive integer:

x^{n}+y^{n}can be rewritten as x^{2a+1}+y^{2a+1}, where a is a positive integer.

1) Base case (a=1)

x^{3}+y^{3}= (x+y)(x^{2}-xy+y^{2}), and (x^{2}-xy+y^{2}) is an integer, so it is divisible.

2) Assumption (a=k)

Assume x^{2k+1}+y^{2k+1}is divisible by (x+y)

3) Next 'step' (a=k+1)

x^{2(k+1)+1}+y^{2(k+1)+1 }=x^{2k+3}+y^{2k+3}

= (x^{2k+1}+y^{2k+1})(x^{2}) - x^{2}y^{2k+1 }+ y^{2k+3 }= (x^{2k+1}+y^{2k+1})(x^{2}) + (x^{2k+1}+y^{2k+1})(y^{2}) - x^{2k+1}y^{2}- x^{2}y^{2k+1 }= (x^{2k+1}+y^{2k+1})(x^{2}) + (x^{2k+1}+y^{2k+1})(y^{2}) - (x+y)(x^{2k}y^{2}) + x^{2k}y^{3}- x^{2}y^{2k+1 }= (x^{2k+1}+y^{2k+1})(x^{2}) + (x^{2k+1}+y^{2k+1})(y^{2}) - (x+y)(x^{2k}y^{2}) - (x+y)(y^{2k}x^{2}) + x^{3}y^{2k }+x^{2k}y^{3}... I tried a bunch more steps/other factors, but nothing seems to be working