Divide using synthetic division $$ ( x^{3}-9x+4 ) \div ( x-5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}5&1&0&-9&4\\& & 5& 25& \color{black}{80} \\ \hline &\color{blue}{1}&\color{blue}{5}&\color{blue}{16}&\color{orangered}{84} \end{array} $$The solution is:
$$ \frac{ x^{3}-9x+4 }{ x-5 } = \color{blue}{x^{2}+5x+16} ~+~ \frac{ \color{red}{ 84 } }{ x-5 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{5}&1&0&-9&4\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}5&\color{orangered}{ 1 }&0&-9&4\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrr}\color{blue}{5}&1&0&-9&4\\& & \color{blue}{5} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 5 } = \color{orangered}{ 5 } $
$$ \begin{array}{c|rrrr}5&1&\color{orangered}{ 0 }&-9&4\\& & \color{orangered}{5} & & \\ \hline &1&\color{orangered}{5}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 5 } = \color{blue}{ 25 } $.
$$ \begin{array}{c|rrrr}\color{blue}{5}&1&0&-9&4\\& & 5& \color{blue}{25} & \\ \hline &1&\color{blue}{5}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -9 } + \color{orangered}{ 25 } = \color{orangered}{ 16 } $
$$ \begin{array}{c|rrrr}5&1&0&\color{orangered}{ -9 }&4\\& & 5& \color{orangered}{25} & \\ \hline &1&5&\color{orangered}{16}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 16 } = \color{blue}{ 80 } $.
$$ \begin{array}{c|rrrr}\color{blue}{5}&1&0&-9&4\\& & 5& 25& \color{blue}{80} \\ \hline &1&5&\color{blue}{16}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 4 } + \color{orangered}{ 80 } = \color{orangered}{ 84 } $
$$ \begin{array}{c|rrrr}5&1&0&-9&\color{orangered}{ 4 }\\& & 5& 25& \color{orangered}{80} \\ \hline &\color{blue}{1}&\color{blue}{5}&\color{blue}{16}&\color{orangered}{84} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{2}+5x+16 } $ with a remainder of $ \color{red}{ 84 } $.