Divide using synthetic division $$ ( x^{3}-5x^{2}-33x-37 ) \div ( x-9 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}9&1&-5&-33&-37\\& & 9& 36& \color{black}{27} \\ \hline &\color{blue}{1}&\color{blue}{4}&\color{blue}{3}&\color{orangered}{-10} \end{array} $$The solution is:
$$ \frac{ x^{3}-5x^{2}-33x-37 }{ x-9 } = \color{blue}{x^{2}+4x+3} \color{red}{~-~} \frac{ \color{red}{ 10 } }{ x-9 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{9}&1&-5&-33&-37\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}9&\color{orangered}{ 1 }&-5&-33&-37\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrr}\color{blue}{9}&1&-5&-33&-37\\& & \color{blue}{9} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -5 } + \color{orangered}{ 9 } = \color{orangered}{ 4 } $
$$ \begin{array}{c|rrrr}9&1&\color{orangered}{ -5 }&-33&-37\\& & \color{orangered}{9} & & \\ \hline &1&\color{orangered}{4}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 9 } \cdot \color{blue}{ 4 } = \color{blue}{ 36 } $.
$$ \begin{array}{c|rrrr}\color{blue}{9}&1&-5&-33&-37\\& & 9& \color{blue}{36} & \\ \hline &1&\color{blue}{4}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -33 } + \color{orangered}{ 36 } = \color{orangered}{ 3 } $
$$ \begin{array}{c|rrrr}9&1&-5&\color{orangered}{ -33 }&-37\\& & 9& \color{orangered}{36} & \\ \hline &1&4&\color{orangered}{3}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 9 } \cdot \color{blue}{ 3 } = \color{blue}{ 27 } $.
$$ \begin{array}{c|rrrr}\color{blue}{9}&1&-5&-33&-37\\& & 9& 36& \color{blue}{27} \\ \hline &1&4&\color{blue}{3}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -37 } + \color{orangered}{ 27 } = \color{orangered}{ -10 } $
$$ \begin{array}{c|rrrr}9&1&-5&-33&\color{orangered}{ -37 }\\& & 9& 36& \color{orangered}{27} \\ \hline &\color{blue}{1}&\color{blue}{4}&\color{blue}{3}&\color{orangered}{-10} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{2}+4x+3 } $ with a remainder of $ \color{red}{ -10 } $.