Answer
$x^{15}+x^{14}+x^{13}+x^{12}+x^{11}+x^{10}+x^{9}+x^{8}+x^{7}+x^{6}+x^{5}+x^{4}+x^{3}+x^{2}+x+1$.
The long division method is preferred.
Work Step by Step
The long division below shows that
$$\begin{aligned}
&(x^{16}-1)\div(x-1)
\\&=
x^{15}+x^{14}+x^{13}+x^{12}+x^{11}+x^{10}+
\\&\phantom{=\,\,\,}x^{9}+x^{8}+x^{7}+x^{6}+x^{5}+x^{4}+x^{3}+
\\&\phantom{=\,\,}x^{2}+x+1
.\end{aligned}
$$
Using $x^{2n}-y^{2n}=(x^n+y^n)(x^n-y^n)$, the given expression, $\frac{x^{16}-1}{x-1}$, is equivalent to
$$\begin{aligned}
&
\frac{(x^{8}+1)(x^{8}-1)}{x-1}
\\&=
\frac{(x^{8}+1)(x^{4}+1)(x^{4}-1)}{x-1}
\\&=
\frac{(x^{8}+1)(x^{4}+1)(x^{2}+1)(x^{2}-1)}{x-1}
\\&=
\frac{(x^{8}+1)(x^{4}+1)(x^{2}+1)(x+1)(x-1)}{x-1}
.\end{aligned}$$Cancelling $x-1$, then the quotient is
$$
(x^{8}+1)(x^{4}+1)(x^{2}+1)(x+1)
.$$ Multiplying all the factors results in
$$\begin{aligned}
&x^{15}+x^{14}+x^{13}+x^{12}+x^{11}+x^{10}+
\\&\phantom{=}x^{9}+x^{8}+x^{7}+x^{6}+x^{5}+x^{4}+x^{3}+
\\&\phantom{=}x^{2}+x+1
.\end{aligned}$$
Hence, both the long division method and factoring method give the same result. The long division method though is preferred since this solution gives a direct approach on how to get the answer.
