Divide using synthetic division $$ ( x^{3}-4x^{2}-23x-32 ) \div ( x-9 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}9&1&-4&-23&-32\\& & 9& 45& \color{black}{198} \\ \hline &\color{blue}{1}&\color{blue}{5}&\color{blue}{22}&\color{orangered}{166} \end{array} $$The solution is:
$$ \frac{ x^{3}-4x^{2}-23x-32 }{ x-9 } = \color{blue}{x^{2}+5x+22} ~+~ \frac{ \color{red}{ 166 } }{ x-9 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{9}&1&-4&-23&-32\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}9&\color{orangered}{ 1 }&-4&-23&-32\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 9 } \cdot \color{blue}{ 1 } = \color{blue}{ 9 } $.
$$ \begin{array}{c|rrrr}\color{blue}{9}&1&-4&-23&-32\\& & \color{blue}{9} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -4 } + \color{orangered}{ 9 } = \color{orangered}{ 5 } $
$$ \begin{array}{c|rrrr}9&1&\color{orangered}{ -4 }&-23&-32\\& & \color{orangered}{9} & & \\ \hline &1&\color{orangered}{5}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 9 } \cdot \color{blue}{ 5 } = \color{blue}{ 45 } $.
$$ \begin{array}{c|rrrr}\color{blue}{9}&1&-4&-23&-32\\& & 9& \color{blue}{45} & \\ \hline &1&\color{blue}{5}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -23 } + \color{orangered}{ 45 } = \color{orangered}{ 22 } $
$$ \begin{array}{c|rrrr}9&1&-4&\color{orangered}{ -23 }&-32\\& & 9& \color{orangered}{45} & \\ \hline &1&5&\color{orangered}{22}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 9 } \cdot \color{blue}{ 22 } = \color{blue}{ 198 } $.
$$ \begin{array}{c|rrrr}\color{blue}{9}&1&-4&-23&-32\\& & 9& 45& \color{blue}{198} \\ \hline &1&5&\color{blue}{22}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -32 } + \color{orangered}{ 198 } = \color{orangered}{ 166 } $
$$ \begin{array}{c|rrrr}9&1&-4&-23&\color{orangered}{ -32 }\\& & 9& 45& \color{orangered}{198} \\ \hline &\color{blue}{1}&\color{blue}{5}&\color{blue}{22}&\color{orangered}{166} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{2}+5x+22 } $ with a remainder of $ \color{red}{ 166 } $.