Divide using synthetic division $$ ( x^{3}+17x^{2}+70x-54 ) \div ( x-9 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}9&1&17&70&-54\\& & 9& 234& \color{black}{2736} \\ \hline &\color{blue}{1}&\color{blue}{26}&\color{blue}{304}&\color{orangered}{2682} \end{array} $$The solution is:
$$ \frac{ x^{3}+17x^{2}+70x-54 }{ x-9 } = \color{blue}{x^{2}+26x+304} ~+~ \frac{ \color{red}{ 2682 } }{ x-9 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{9}&1&17&70&-54\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}9&\color{orangered}{ 1 }&17&70&-54\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 9 } \cdot \color{blue}{ 1 } = \color{blue}{ 9 } $.
$$ \begin{array}{c|rrrr}\color{blue}{9}&1&17&70&-54\\& & \color{blue}{9} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 17 } + \color{orangered}{ 9 } = \color{orangered}{ 26 } $
$$ \begin{array}{c|rrrr}9&1&\color{orangered}{ 17 }&70&-54\\& & \color{orangered}{9} & & \\ \hline &1&\color{orangered}{26}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 9 } \cdot \color{blue}{ 26 } = \color{blue}{ 234 } $.
$$ \begin{array}{c|rrrr}\color{blue}{9}&1&17&70&-54\\& & 9& \color{blue}{234} & \\ \hline &1&\color{blue}{26}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 70 } + \color{orangered}{ 234 } = \color{orangered}{ 304 } $
$$ \begin{array}{c|rrrr}9&1&17&\color{orangered}{ 70 }&-54\\& & 9& \color{orangered}{234} & \\ \hline &1&26&\color{orangered}{304}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 9 } \cdot \color{blue}{ 304 } = \color{blue}{ 2736 } $.
$$ \begin{array}{c|rrrr}\color{blue}{9}&1&17&70&-54\\& & 9& 234& \color{blue}{2736} \\ \hline &1&26&\color{blue}{304}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -54 } + \color{orangered}{ 2736 } = \color{orangered}{ 2682 } $
$$ \begin{array}{c|rrrr}9&1&17&70&\color{orangered}{ -54 }\\& & 9& 234& \color{orangered}{2736} \\ \hline &\color{blue}{1}&\color{blue}{26}&\color{blue}{304}&\color{orangered}{2682} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{2}+26x+304 } $ with a remainder of $ \color{red}{ 2682 } $.