Divide using synthetic division $$ ( 2x^{4}-7x^{3}-17x^{2}+58x-24 ) \div ( x-9 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}9&2&-7&-17&58&-24\\& & 18& 99& 738& \color{black}{7164} \\ \hline &\color{blue}{2}&\color{blue}{11}&\color{blue}{82}&\color{blue}{796}&\color{orangered}{7140} \end{array} $$The solution is:
$$ \frac{ 2x^{4}-7x^{3}-17x^{2}+58x-24 }{ x-9 } = \color{blue}{2x^{3}+11x^{2}+82x+796} ~+~ \frac{ \color{red}{ 7140 } }{ x-9 } $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{9}&2&-7&-17&58&-24\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}9&\color{orangered}{ 2 }&-7&-17&58&-24\\& & & & & \\ \hline &\color{orangered}{2}&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{9}&2&-7&-17&58&-24\\& & \color{blue}{18} & & & \\ \hline &\color{blue}{2}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -7 } + \color{orangered}{ 18 } = \color{orangered}{ 11 } $
$$ \begin{array}{c|rrrrr}9&2&\color{orangered}{ -7 }&-17&58&-24\\& & \color{orangered}{18} & & & \\ \hline &2&\color{orangered}{11}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 9 } \cdot \color{blue}{ 11 } = \color{blue}{ 99 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{9}&2&-7&-17&58&-24\\& & 18& \color{blue}{99} & & \\ \hline &2&\color{blue}{11}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -17 } + \color{orangered}{ 99 } = \color{orangered}{ 82 } $
$$ \begin{array}{c|rrrrr}9&2&-7&\color{orangered}{ -17 }&58&-24\\& & 18& \color{orangered}{99} & & \\ \hline &2&11&\color{orangered}{82}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 9 } \cdot \color{blue}{ 82 } = \color{blue}{ 738 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{9}&2&-7&-17&58&-24\\& & 18& 99& \color{blue}{738} & \\ \hline &2&11&\color{blue}{82}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 58 } + \color{orangered}{ 738 } = \color{orangered}{ 796 } $
$$ \begin{array}{c|rrrrr}9&2&-7&-17&\color{orangered}{ 58 }&-24\\& & 18& 99& \color{orangered}{738} & \\ \hline &2&11&82&\color{orangered}{796}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 9 } \cdot \color{blue}{ 796 } = \color{blue}{ 7164 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{9}&2&-7&-17&58&-24\\& & 18& 99& 738& \color{blue}{7164} \\ \hline &2&11&82&\color{blue}{796}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -24 } + \color{orangered}{ 7164 } = \color{orangered}{ 7140 } $
$$ \begin{array}{c|rrrrr}9&2&-7&-17&58&\color{orangered}{ -24 }\\& & 18& 99& 738& \color{orangered}{7164} \\ \hline &\color{blue}{2}&\color{blue}{11}&\color{blue}{82}&\color{blue}{796}&\color{orangered}{7140} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x^{3}+11x^{2}+82x+796 } $ with a remainder of $ \color{red}{ 7140 } $.