Use factor theorem to prove that (x +a)is a factor of $(x^{n} + a^{n})$ for any odd positive integer n.

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Solution

ANSWER:
Let $f (x) = x^{n} + a^{n}$
Putting x = −a in f(x), we get
$f (- a) = (- a)^{n} + a^{n}$
If n is any odd positive integer, then
$f (- a) = (- a)^{n} + a^{n} = - a^{n} + a^{n} = 0$
Therefore, by factor theorem, (x + a) is a factor of $(x^{n} + a^{n})$ for any odd positive integer.

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