Aligning \vrule in tables

Question

I'm making tables for displaying the method to represent how factorial polynomials are derived by using a tabular method:

As can be seen here, the table has partially ruled vertical lines, trying to replicate this for my assignment using \vrule has given me some issues:

As can be seen, the \vrule do not align, any help is greatly appreciated

MWE

\begin{tabular}{cc|cc|ccc}
   1& &2& &6& &\vrule-8\\
   & & & &2& &\vrule\\
 \cline{3-6}
   2& &2& &8& &\\
 \cline{3-4}
\end{tabular}

Note: apologies for atrocious formatting, I wrote this on my phone in a hurry.

Answer 1

Example that uses array for math mode in the cells (minus sign, digits), vertical tabular lines, which can be overwritten by \multicolumn, and \cline:

\documentclass{article}

\begin{document}
$\begin{array}{r|r|r|r}
  1 & 2 & 6 & -8 \\
  & & 2 & \\
  \cline{2-3}
  2 & 2 & \multicolumn{1}{r}{8} \\
  & \\
  \cline{2-2}
\end{array}$
\end{document}

Also package siunitx can be used to properly align the numbers, but adding some space to the right by place for phantom digits. (The reason for the additional space to the right eludes me.)

\documentclass{article}
\usepackage{siunitx}

\begin{document}
$\sisetup{table-format=1.1}
\begin{array}{S|S|S|S[table-format=-1]}
  1 & 2 & 6 & -8 \\
  & & 2 & \\
  \cline{2-3}
  2 & 2 & \multicolumn{1}{S}{8} \\
  & \\
  \cline{2-2}
\end{array}$
\end{document}