Divide the following by the synthetic division method: $p(x) = x^{4} - 1$ by $x - 1$.

(N/A) To divide $p(x) = x^{4} - 1$ by $x - 1$ using synthetic division:
$1$. Write the dividend $p(x)$ in standard form including all missing terms with zero coefficients: $p(x) = 1x^{4} + 0x^{3} + 0x^{2} + 0x - 1$.
$2$. The divisor is $x - 1$,so set $x - 1 = 0$,which gives $x = 1$. Use $1$ for the synthetic division.
$3$. Set up the synthetic division table:
$\begin{array}{c|ccccc} 1 & 1 & 0 & 0 & 0 & -1 \\ & & 1 & 1 & 1 & 1 \\ \hline & 1 & 1 & 1 & 1 & 0 \end{array}$
$4$. The bottom row represents the coefficients of the quotient and the remainder. The last value is the remainder.
Thus,the quotient is $q(x) = x^{3} + x^{2} + x + 1$ and the remainder is $r(x) = 0$.