Divide using synthetic division $$ ( 2x^{4}-3x^{3}-17x^{2}+16 ) \div ( x-9 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}9&2&-3&-17&0&16\\& & 18& 135& 1062& \color{black}{9558} \\ \hline &\color{blue}{2}&\color{blue}{15}&\color{blue}{118}&\color{blue}{1062}&\color{orangered}{9574} \end{array} $$The solution is:
$$ \frac{ 2x^{4}-3x^{3}-17x^{2}+16 }{ x-9 } = \color{blue}{2x^{3}+15x^{2}+118x+1062} ~+~ \frac{ \color{red}{ 9574 } }{ x-9 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -9 = 0 $ ( $ x = \color{blue}{ 9 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{9}&2&-3&-17&0&16\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}9&\color{orangered}{ 2 }&-3&-17&0&16\\& & & & & \\ \hline &\color{orangered}{2}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 9 } \cdot \color{blue}{ 2 } = \color{blue}{ 18 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{9}&2&-3&-17&0&16\\& & \color{blue}{18} & & & \\ \hline &\color{blue}{2}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ 18 } = \color{orangered}{ 15 } $
$$ \begin{array}{c|rrrrr}9&2&\color{orangered}{ -3 }&-17&0&16\\& & \color{orangered}{18} & & & \\ \hline &2&\color{orangered}{15}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 9 } \cdot \color{blue}{ 15 } = \color{blue}{ 135 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{9}&2&-3&-17&0&16\\& & 18& \color{blue}{135} & & \\ \hline &2&\color{blue}{15}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -17 } + \color{orangered}{ 135 } = \color{orangered}{ 118 } $
$$ \begin{array}{c|rrrrr}9&2&-3&\color{orangered}{ -17 }&0&16\\& & 18& \color{orangered}{135} & & \\ \hline &2&15&\color{orangered}{118}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 9 } \cdot \color{blue}{ 118 } = \color{blue}{ 1062 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{9}&2&-3&-17&0&16\\& & 18& 135& \color{blue}{1062} & \\ \hline &2&15&\color{blue}{118}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 1062 } = \color{orangered}{ 1062 } $
$$ \begin{array}{c|rrrrr}9&2&-3&-17&\color{orangered}{ 0 }&16\\& & 18& 135& \color{orangered}{1062} & \\ \hline &2&15&118&\color{orangered}{1062}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 9 } \cdot \color{blue}{ 1062 } = \color{blue}{ 9558 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{9}&2&-3&-17&0&16\\& & 18& 135& 1062& \color{blue}{9558} \\ \hline &2&15&118&\color{blue}{1062}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 16 } + \color{orangered}{ 9558 } = \color{orangered}{ 9574 } $
$$ \begin{array}{c|rrrrr}9&2&-3&-17&0&\color{orangered}{ 16 }\\& & 18& 135& 1062& \color{orangered}{9558} \\ \hline &\color{blue}{2}&\color{blue}{15}&\color{blue}{118}&\color{blue}{1062}&\color{orangered}{9574} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x^{3}+15x^{2}+118x+1062 } $ with a remainder of $ \color{red}{ 9574 } $.