Question

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

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Answer 1

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Answer 2

Answer:

Step-by-step explanation:

Depending on where the variables live, this might not be true. But the flavour of the question brings normal secondary math to mind so we will say real numbers.

Let f(x)=x^n+a^n

. The Factor Theorem says to examine f(−a). If its value is 0, x+a is a factor of f(x)

If n

is odd, f(−a)=(−a)^n+a^n=−a^n+a^n=0

If n

is even, f(−a)=(−a)^n+a^n=a^n+a^n which is not 0.

or

Using the Polynomial Remainder Theorem, when p(x) is divided by x−a,a being any number, the remainder is equal to p(a).

Now,here p(x)=x^n+a^n

we can write x+a as x−(−a), and so, the remainder will be equal to p(−a)

This comes out to be −a^n+a^n    which will be zero for all odd n

Thus the remainder will be zero and we get the required result.