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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.
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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