Question

4. Use factor theorem to verify that (x + a) is a factor of x + an for any odd positive integer.
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Answer 1

$\huge{\textsf{\textbf{Answer:}}}$

Let f(x)=x n +an .

In order to prove that x+a is a factor of f(x) for any odd positive integer n, it is sufficient to show that f(−a)=0.

f(−a)=(−a)

n +a n =(−1) n a

n +a n

f(−a)=(−1+1)a n

[ n is odd positive integer ]

f(−a)=0×a

n =0

Hence, x+a is a factor of x

n +a n

, when n is an odd positive integer.