Review question

# When is $x^n + 1$ divisible by $x + 1$? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R7108

## Solution

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

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