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Question

Use factor theorem to prove that (x + a) is a factor of (xn + an) for any odd positive integer.

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Solution


Let f(x) = xn + an

Putting x = −a in f(x), we get

f(−a) = (−a)n + an

If n is any odd positive integer, then

f(−a) = (−a)n + an = −an + an = 0

Therefore, by factor theorem, (x + a) is a factor of (xn + an) for any odd positive integer.

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