Explain why a sum of odd powers is factorable, but not a sum of even powers.

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`x^1+y^1=(x+y-y)^1+y^1=(x+y)^1+(-y)^1+y^1`

`=(x+y)-y+y=(x+y)`

`x^2+y^2=(x+y-y)^2+y^2=(x+y)^2-2xy+(-y)^2+y^2`

`=(x+y)^2-2xy+y^2+y^2=(x+y)^2-2xy+2y^2`

`=(x+y)^2-2y(x-y)`

No common factor.

`x^3+y^3=(x+y-y)^3+y^3`

`=(x+y)^3-3(x+y)y(x+y-y)+(-y)^3+y^3`

`=(x+y)^3-3xy(x+y)-y^3+y^3`

`=(x+y)^3-3xy(x+y)`

In this expression there is common factor (x+y)

`=(x+y)((x+y)^2-3xy)=(x+y)(x^2+y^2-xy)`

`` Thus we can explain this

1. If it is even power i.e. `y^(2m)=(-y)^(2m)`

2. If it is odd power i.e. `y^(2m+1)!=(-y)^(2m+1)`

This is the reason why sum of even powers not factorable.

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