Divide using synthetic division $$ ( x^{4}-6x^{3}-33x^{2}+46x+75 ) \div ( x-9 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}9&1&-6&-33&46&75\\& & 9& 27& -54& \color{black}{-72} \\ \hline &\color{blue}{1}&\color{blue}{3}&\color{blue}{-6}&\color{blue}{-8}&\color{orangered}{3} \end{array} $$The solution is:
$$ \frac{ x^{4}-6x^{3}-33x^{2}+46x+75 }{ x-9 } = \color{blue}{x^{3}+3x^{2}-6x-8} ~+~ \frac{ \color{red}{ 3 } }{ x-9 } $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{9}&1&-6&-33&46&75\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}9&\color{orangered}{ 1 }&-6&-33&46&75\\& & & & & \\ \hline &\color{orangered}{1}&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{9}&1&-6&-33&46&75\\& & \color{blue}{9} & & & \\ \hline &\color{blue}{1}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -6 } + \color{orangered}{ 9 } = \color{orangered}{ 3 } $
$$ \begin{array}{c|rrrrr}9&1&\color{orangered}{ -6 }&-33&46&75\\& & \color{orangered}{9} & & & \\ \hline &1&\color{orangered}{3}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 9 } \cdot \color{blue}{ 3 } = \color{blue}{ 27 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{9}&1&-6&-33&46&75\\& & 9& \color{blue}{27} & & \\ \hline &1&\color{blue}{3}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -33 } + \color{orangered}{ 27 } = \color{orangered}{ -6 } $
$$ \begin{array}{c|rrrrr}9&1&-6&\color{orangered}{ -33 }&46&75\\& & 9& \color{orangered}{27} & & \\ \hline &1&3&\color{orangered}{-6}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 9 } \cdot \color{blue}{ \left( -6 \right) } = \color{blue}{ -54 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{9}&1&-6&-33&46&75\\& & 9& 27& \color{blue}{-54} & \\ \hline &1&3&\color{blue}{-6}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 46 } + \color{orangered}{ \left( -54 \right) } = \color{orangered}{ -8 } $
$$ \begin{array}{c|rrrrr}9&1&-6&-33&\color{orangered}{ 46 }&75\\& & 9& 27& \color{orangered}{-54} & \\ \hline &1&3&-6&\color{orangered}{-8}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 9 } \cdot \color{blue}{ \left( -8 \right) } = \color{blue}{ -72 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{9}&1&-6&-33&46&75\\& & 9& 27& -54& \color{blue}{-72} \\ \hline &1&3&-6&\color{blue}{-8}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 75 } + \color{orangered}{ \left( -72 \right) } = \color{orangered}{ 3 } $
$$ \begin{array}{c|rrrrr}9&1&-6&-33&46&\color{orangered}{ 75 }\\& & 9& 27& -54& \color{orangered}{-72} \\ \hline &\color{blue}{1}&\color{blue}{3}&\color{blue}{-6}&\color{blue}{-8}&\color{orangered}{3} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{3}+3x^{2}-6x-8 } $ with a remainder of $ \color{red}{ 3 } $.