Divide using synthetic division $$ ( x^{3}-5x^{2}+9 ) \div ( x-6 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}6&1&-5&0&9\\& & 6& 6& \color{black}{36} \\ \hline &\color{blue}{1}&\color{blue}{1}&\color{blue}{6}&\color{orangered}{45} \end{array} $$The solution is:
$$ \frac{ x^{3}-5x^{2}+9 }{ x-6 } = \color{blue}{x^{2}+x+6} ~+~ \frac{ \color{red}{ 45 } }{ x-6 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -6 = 0 $ ( $ x = \color{blue}{ 6 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{6}&1&-5&0&9\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}6&\color{orangered}{ 1 }&-5&0&9\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ 1 } = \color{blue}{ 6 } $.
$$ \begin{array}{c|rrrr}\color{blue}{6}&1&-5&0&9\\& & \color{blue}{6} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -5 } + \color{orangered}{ 6 } = \color{orangered}{ 1 } $
$$ \begin{array}{c|rrrr}6&1&\color{orangered}{ -5 }&0&9\\& & \color{orangered}{6} & & \\ \hline &1&\color{orangered}{1}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ 1 } = \color{blue}{ 6 } $.
$$ \begin{array}{c|rrrr}\color{blue}{6}&1&-5&0&9\\& & 6& \color{blue}{6} & \\ \hline &1&\color{blue}{1}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 6 } = \color{orangered}{ 6 } $
$$ \begin{array}{c|rrrr}6&1&-5&\color{orangered}{ 0 }&9\\& & 6& \color{orangered}{6} & \\ \hline &1&1&\color{orangered}{6}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ 6 } = \color{blue}{ 36 } $.
$$ \begin{array}{c|rrrr}\color{blue}{6}&1&-5&0&9\\& & 6& 6& \color{blue}{36} \\ \hline &1&1&\color{blue}{6}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 9 } + \color{orangered}{ 36 } = \color{orangered}{ 45 } $
$$ \begin{array}{c|rrrr}6&1&-5&0&\color{orangered}{ 9 }\\& & 6& 6& \color{orangered}{36} \\ \hline &\color{blue}{1}&\color{blue}{1}&\color{blue}{6}&\color{orangered}{45} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{2}+x+6 } $ with a remainder of $ \color{red}{ 45 } $.