Divide using synthetic division $$ ( 2x^{4}-3x^{3}-17x^{2}+16 ) \div ( x-1 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}1&2&-3&-17&0&16\\& & 2& -1& -18& \color{black}{-18} \\ \hline &\color{blue}{2}&\color{blue}{-1}&\color{blue}{-18}&\color{blue}{-18}&\color{orangered}{-2} \end{array} $$The solution is:
$$ \frac{ 2x^{4}-3x^{3}-17x^{2}+16 }{ x-1 } = \color{blue}{2x^{3}-x^{2}-18x-18} \color{red}{~-~} \frac{ \color{red}{ 2 } }{ x-1 } $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{1}&2&-3&-17&0&16\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}1&\color{orangered}{ 2 }&-3&-17&0&16\\& & & & & \\ \hline &\color{orangered}{2}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 2 } = \color{blue}{ 2 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{1}&2&-3&-17&0&16\\& & \color{blue}{2} & & & \\ \hline &\color{blue}{2}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ 2 } = \color{orangered}{ -1 } $
$$ \begin{array}{c|rrrrr}1&2&\color{orangered}{ -3 }&-17&0&16\\& & \color{orangered}{2} & & & \\ \hline &2&\color{orangered}{-1}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ -1 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{1}&2&-3&-17&0&16\\& & 2& \color{blue}{-1} & & \\ \hline &2&\color{blue}{-1}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -17 } + \color{orangered}{ \left( -1 \right) } = \color{orangered}{ -18 } $
$$ \begin{array}{c|rrrrr}1&2&-3&\color{orangered}{ -17 }&0&16\\& & 2& \color{orangered}{-1} & & \\ \hline &2&-1&\color{orangered}{-18}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ \left( -18 \right) } = \color{blue}{ -18 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{1}&2&-3&-17&0&16\\& & 2& -1& \color{blue}{-18} & \\ \hline &2&-1&\color{blue}{-18}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -18 \right) } = \color{orangered}{ -18 } $
$$ \begin{array}{c|rrrrr}1&2&-3&-17&\color{orangered}{ 0 }&16\\& & 2& -1& \color{orangered}{-18} & \\ \hline &2&-1&-18&\color{orangered}{-18}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ \left( -18 \right) } = \color{blue}{ -18 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{1}&2&-3&-17&0&16\\& & 2& -1& -18& \color{blue}{-18} \\ \hline &2&-1&-18&\color{blue}{-18}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 16 } + \color{orangered}{ \left( -18 \right) } = \color{orangered}{ -2 } $
$$ \begin{array}{c|rrrrr}1&2&-3&-17&0&\color{orangered}{ 16 }\\& & 2& -1& -18& \color{orangered}{-18} \\ \hline &\color{blue}{2}&\color{blue}{-1}&\color{blue}{-18}&\color{blue}{-18}&\color{orangered}{-2} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x^{3}-x^{2}-18x-18 } $ with a remainder of $ \color{red}{ -2 } $.