# Let us show that if n is any positive integer (even or odd), x – y is a factor of the polynomial xn – yn.

Formula used.

If f(x) is a polynomial with degree n

Then (x – a) is a factor of f(x) if f(a) = 0

Dividend = Divisor × Quotient + Remainder

If (x – y) is a factor of xn – yn

Then

We have to prove

On dividing xn – yn by (x – y) Remainder gets 0

When n = 1

(x – y) becomes factor of (x1 – y1)

Hence,

x – y = 0

x = y

Suppose on dividing xn – yn with x – y we get Remainder R

xn – yn = (x – y) × Quotient + R

Putting x = y

yn – yn = (y – y) × Quotient + R

0 = 0 + R

R = 0

For every value n (x – y) is a factor of xn – yn

Conclusion.

For every value n (x – y) is a factor of xn – yn

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