Q. 95.0( 5 Votes )

# Let us show that if n is any positive odd integer, then x + y is a factor of x^{n} + y^{n}.

Answer :

Let us suppose, if x^{n} + y^{n} is divided by x + y, the quotient is Q and remainder without x is R.

Dividend = Divisor × Quotient + Remainder

⸫ x^{n} + y^{n} = (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} + y^{n} = (-y + y) × Q + R

Now, as n is odd, we get that (-y)^{n} = - y^{n}

so, we get,

- y^{n} + y^{n} = 0 × Q + R

0 = R

⸫ R = 0

⸫ (x + y) is a factor of the polynomial x^{n} + y^{n}, when n is an odd positive integer.

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