Divide using synthetic division $$ ( x^{4}+16x^{3}+65x^{2}+23x+50 ) \div ( x-9 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}9&1&16&65&23&50\\& & 9& 225& 2610& \color{black}{23697} \\ \hline &\color{blue}{1}&\color{blue}{25}&\color{blue}{290}&\color{blue}{2633}&\color{orangered}{23747} \end{array} $$The solution is:
$$ \frac{ x^{4}+16x^{3}+65x^{2}+23x+50 }{ x-9 } = \color{blue}{x^{3}+25x^{2}+290x+2633} ~+~ \frac{ \color{red}{ 23747 } }{ x-9 } $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{9}&1&16&65&23&50\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}9&\color{orangered}{ 1 }&16&65&23&50\\& & & & & \\ \hline &\color{orangered}{1}&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{9}&1&16&65&23&50\\& & \color{blue}{9} & & & \\ \hline &\color{blue}{1}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 16 } + \color{orangered}{ 9 } = \color{orangered}{ 25 } $
$$ \begin{array}{c|rrrrr}9&1&\color{orangered}{ 16 }&65&23&50\\& & \color{orangered}{9} & & & \\ \hline &1&\color{orangered}{25}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 9 } \cdot \color{blue}{ 25 } = \color{blue}{ 225 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{9}&1&16&65&23&50\\& & 9& \color{blue}{225} & & \\ \hline &1&\color{blue}{25}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 65 } + \color{orangered}{ 225 } = \color{orangered}{ 290 } $
$$ \begin{array}{c|rrrrr}9&1&16&\color{orangered}{ 65 }&23&50\\& & 9& \color{orangered}{225} & & \\ \hline &1&25&\color{orangered}{290}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 9 } \cdot \color{blue}{ 290 } = \color{blue}{ 2610 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{9}&1&16&65&23&50\\& & 9& 225& \color{blue}{2610} & \\ \hline &1&25&\color{blue}{290}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 23 } + \color{orangered}{ 2610 } = \color{orangered}{ 2633 } $
$$ \begin{array}{c|rrrrr}9&1&16&65&\color{orangered}{ 23 }&50\\& & 9& 225& \color{orangered}{2610} & \\ \hline &1&25&290&\color{orangered}{2633}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 9 } \cdot \color{blue}{ 2633 } = \color{blue}{ 23697 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{9}&1&16&65&23&50\\& & 9& 225& 2610& \color{blue}{23697} \\ \hline &1&25&290&\color{blue}{2633}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 50 } + \color{orangered}{ 23697 } = \color{orangered}{ 23747 } $
$$ \begin{array}{c|rrrrr}9&1&16&65&23&\color{orangered}{ 50 }\\& & 9& 225& 2610& \color{orangered}{23697} \\ \hline &\color{blue}{1}&\color{blue}{25}&\color{blue}{290}&\color{blue}{2633}&\color{orangered}{23747} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{3}+25x^{2}+290x+2633 } $ with a remainder of $ \color{red}{ 23747 } $.