Question

$(x+1)$ is a factors of $x^n+1$ only if:

• A. $n$ is an odd integer
• B. $n$ is an even integer
• C. $n$ is a negative integer
• D. $n$ is a positive integer

Answer 1

The correct option is A $n$ is an odd integer

We have,

Let, $n$ is odd number.

Then, $x^n+1$ can be factorized as $(x+1)(1-x+x^2=x^3+.....x^{n-1})$.

Given, $(x+1)$ is a factor.

Take $p\left(x\right)=x^{n+1}$.

When $n$ is odd, we get, $p(-1)=-1+1$$=0$.

Therefore, $(x+1)$ is a factor of $p\left(x\right)$.

Hence, option $A$ is correct.