When is $x^n + 1$ divisible by $x + 1$?

When is $x^n + 1$ divisible by $x + 1$?

Question

Show that $x^n + 1$ is divisible by $x + 1$ when, and only when, $n$ is an odd integer.

Suggestion

We are being asked when a polynomial $p(x)$ is divisible by $x-a$.

Could we use the factor theorem here?

Solution

By the factor theorem, given a polynomial $p(x)$ with real coefficients and some real number $a$, we know that $x-a$ is a factor of $p(x)$ if and only if $p(a) = 0$.

If we let

$p(x) = x^n + 1$

and

$a = -1$

, by the factor theorem we have that

$$\begin{align*} \text{}x^n + 1\text{ is divisible by }x + 1\text{} &\iff (-1)^n + 1 = 0 \\ &\iff (-1)^n = -1 \\ &\iff \text{}n\text{ is an odd integer}. \end{align*}$$