Divide using synthetic division $$ ( x^{3}+3x^{2}-4x+6 ) \div ( x-5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}5&1&3&-4&6\\& & 5& 40& \color{black}{180} \\ \hline &\color{blue}{1}&\color{blue}{8}&\color{blue}{36}&\color{orangered}{186} \end{array} $$The solution is:
$$ \frac{ x^{3}+3x^{2}-4x+6 }{ x-5 } = \color{blue}{x^{2}+8x+36} ~+~ \frac{ \color{red}{ 186 } }{ x-5 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -5 = 0 $ ( $ x = \color{blue}{ 5 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{5}&1&3&-4&6\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}5&\color{orangered}{ 1 }&3&-4&6\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 1 } = \color{blue}{ 5 } $.
$$ \begin{array}{c|rrrr}\color{blue}{5}&1&3&-4&6\\& & \color{blue}{5} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ 5 } = \color{orangered}{ 8 } $
$$ \begin{array}{c|rrrr}5&1&\color{orangered}{ 3 }&-4&6\\& & \color{orangered}{5} & & \\ \hline &1&\color{orangered}{8}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 8 } = \color{blue}{ 40 } $.
$$ \begin{array}{c|rrrr}\color{blue}{5}&1&3&-4&6\\& & 5& \color{blue}{40} & \\ \hline &1&\color{blue}{8}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -4 } + \color{orangered}{ 40 } = \color{orangered}{ 36 } $
$$ \begin{array}{c|rrrr}5&1&3&\color{orangered}{ -4 }&6\\& & 5& \color{orangered}{40} & \\ \hline &1&8&\color{orangered}{36}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 36 } = \color{blue}{ 180 } $.
$$ \begin{array}{c|rrrr}\color{blue}{5}&1&3&-4&6\\& & 5& 40& \color{blue}{180} \\ \hline &1&8&\color{blue}{36}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 6 } + \color{orangered}{ 180 } = \color{orangered}{ 186 } $
$$ \begin{array}{c|rrrr}5&1&3&-4&\color{orangered}{ 6 }\\& & 5& 40& \color{orangered}{180} \\ \hline &\color{blue}{1}&\color{blue}{8}&\color{blue}{36}&\color{orangered}{186} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{2}+8x+36 } $ with a remainder of $ \color{red}{ 186 } $.