Divide using synthetic division $$ ( x^{4}-18x^{3}+31x^{2}+18x-32 ) \div ( x-1 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}1&1&-18&31&18&-32\\& & 1& -17& 14& \color{black}{32} \\ \hline &\color{blue}{1}&\color{blue}{-17}&\color{blue}{14}&\color{blue}{32}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ x^{4}-18x^{3}+31x^{2}+18x-32 }{ x-1 } = \color{blue}{x^{3}-17x^{2}+14x+32} $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -1 = 0 $ ( $ x = \color{blue}{ 1 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{1}&1&-18&31&18&-32\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}1&\color{orangered}{ 1 }&-18&31&18&-32\\& & & & & \\ \hline &\color{orangered}{1}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 1 } = \color{blue}{ 1 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{1}&1&-18&31&18&-32\\& & \color{blue}{1} & & & \\ \hline &\color{blue}{1}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -18 } + \color{orangered}{ 1 } = \color{orangered}{ -17 } $
$$ \begin{array}{c|rrrrr}1&1&\color{orangered}{ -18 }&31&18&-32\\& & \color{orangered}{1} & & & \\ \hline &1&\color{orangered}{-17}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ \left( -17 \right) } = \color{blue}{ -17 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{1}&1&-18&31&18&-32\\& & 1& \color{blue}{-17} & & \\ \hline &1&\color{blue}{-17}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 31 } + \color{orangered}{ \left( -17 \right) } = \color{orangered}{ 14 } $
$$ \begin{array}{c|rrrrr}1&1&-18&\color{orangered}{ 31 }&18&-32\\& & 1& \color{orangered}{-17} & & \\ \hline &1&-17&\color{orangered}{14}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 14 } = \color{blue}{ 14 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{1}&1&-18&31&18&-32\\& & 1& -17& \color{blue}{14} & \\ \hline &1&-17&\color{blue}{14}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 18 } + \color{orangered}{ 14 } = \color{orangered}{ 32 } $
$$ \begin{array}{c|rrrrr}1&1&-18&31&\color{orangered}{ 18 }&-32\\& & 1& -17& \color{orangered}{14} & \\ \hline &1&-17&14&\color{orangered}{32}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 32 } = \color{blue}{ 32 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{1}&1&-18&31&18&-32\\& & 1& -17& 14& \color{blue}{32} \\ \hline &1&-17&14&\color{blue}{32}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -32 } + \color{orangered}{ 32 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrr}1&1&-18&31&18&\color{orangered}{ -32 }\\& & 1& -17& 14& \color{orangered}{32} \\ \hline &\color{blue}{1}&\color{blue}{-17}&\color{blue}{14}&\color{blue}{32}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{3}-17x^{2}+14x+32 } $ with a remainder of $ \color{red}{ 0 } $.