Let us show that if n be any positive integer (even or odd), the x – y never be a factor of the polynomial xⁿ + yⁿ.

Let us suppose, if xⁿ + yⁿ is divided by x - y, the quotient is Q and remainder without x is R.

Dividend = Divisor × Quotient + Remainder

⸫ xⁿ + yⁿ = (x - y) × Q + R ……. [This is an identity]

Since x does not belong to the remainder R, the value of R will not change for any value of x.

So, in the above identity, putting (y) for x, we get:

(y)ⁿ + yⁿ = (y - y) × Q + R

2(yⁿ ) = 0 + R

2(yⁿ ) = R

Since, value of R is not “0”, we can say that x – y is not a factor of the polynomial xⁿ + yⁿ

⸫ we can say that (x – y) can never be a factor of the polynomial xⁿ + yⁿ.