Divide using synthetic division $$ ( x^{3}+9x^{2}+13x+35 ) \div ( x-5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}5&1&9&13&35\\& & 5& 70& \color{black}{415} \\ \hline &\color{blue}{1}&\color{blue}{14}&\color{blue}{83}&\color{orangered}{450} \end{array} $$The solution is:
$$ \frac{ x^{3}+9x^{2}+13x+35 }{ x-5 } = \color{blue}{x^{2}+14x+83} ~+~ \frac{ \color{red}{ 450 } }{ x-5 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{5}&1&9&13&35\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}5&\color{orangered}{ 1 }&9&13&35\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrr}\color{blue}{5}&1&9&13&35\\& & \color{blue}{5} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 9 } + \color{orangered}{ 5 } = \color{orangered}{ 14 } $
$$ \begin{array}{c|rrrr}5&1&\color{orangered}{ 9 }&13&35\\& & \color{orangered}{5} & & \\ \hline &1&\color{orangered}{14}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 14 } = \color{blue}{ 70 } $.
$$ \begin{array}{c|rrrr}\color{blue}{5}&1&9&13&35\\& & 5& \color{blue}{70} & \\ \hline &1&\color{blue}{14}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 13 } + \color{orangered}{ 70 } = \color{orangered}{ 83 } $
$$ \begin{array}{c|rrrr}5&1&9&\color{orangered}{ 13 }&35\\& & 5& \color{orangered}{70} & \\ \hline &1&14&\color{orangered}{83}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 83 } = \color{blue}{ 415 } $.
$$ \begin{array}{c|rrrr}\color{blue}{5}&1&9&13&35\\& & 5& 70& \color{blue}{415} \\ \hline &1&14&\color{blue}{83}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 35 } + \color{orangered}{ 415 } = \color{orangered}{ 450 } $
$$ \begin{array}{c|rrrr}5&1&9&13&\color{orangered}{ 35 }\\& & 5& 70& \color{orangered}{415} \\ \hline &\color{blue}{1}&\color{blue}{14}&\color{blue}{83}&\color{orangered}{450} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{2}+14x+83 } $ with a remainder of $ \color{red}{ 450 } $.