Divide using synthetic division $$ ( 9x^{5}-x+24 ) \div ( 3x-5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}5&9&0&0&0&-1&24\\& & 45& 225& 1125& 5625& \color{black}{28120} \\ \hline &\color{blue}{9}&\color{blue}{45}&\color{blue}{225}&\color{blue}{1125}&\color{blue}{5624}&\color{orangered}{28144} \end{array} $$The solution is:
$$ \frac{ 9x^{5}-x+24 }{ x-5 } = \color{blue}{9x^{4}+45x^{3}+225x^{2}+1125x+5624} ~+~ \frac{ \color{red}{ 28144 } }{ x-5 } $$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.
$$ \begin{array}{c|rrrrrr}\color{blue}{5}&9&0&0&0&-1&24\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}5&\color{orangered}{ 9 }&0&0&0&-1&24\\& & & & & & \\ \hline &\color{orangered}{9}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 9 } = \color{blue}{ 45 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{5}&9&0&0&0&-1&24\\& & \color{blue}{45} & & & & \\ \hline &\color{blue}{9}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 45 } = \color{orangered}{ 45 } $
$$ \begin{array}{c|rrrrrr}5&9&\color{orangered}{ 0 }&0&0&-1&24\\& & \color{orangered}{45} & & & & \\ \hline &9&\color{orangered}{45}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 45 } = \color{blue}{ 225 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{5}&9&0&0&0&-1&24\\& & 45& \color{blue}{225} & & & \\ \hline &9&\color{blue}{45}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 225 } = \color{orangered}{ 225 } $
$$ \begin{array}{c|rrrrrr}5&9&0&\color{orangered}{ 0 }&0&-1&24\\& & 45& \color{orangered}{225} & & & \\ \hline &9&45&\color{orangered}{225}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 225 } = \color{blue}{ 1125 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{5}&9&0&0&0&-1&24\\& & 45& 225& \color{blue}{1125} & & \\ \hline &9&45&\color{blue}{225}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 1125 } = \color{orangered}{ 1125 } $
$$ \begin{array}{c|rrrrrr}5&9&0&0&\color{orangered}{ 0 }&-1&24\\& & 45& 225& \color{orangered}{1125} & & \\ \hline &9&45&225&\color{orangered}{1125}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 1125 } = \color{blue}{ 5625 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{5}&9&0&0&0&-1&24\\& & 45& 225& 1125& \color{blue}{5625} & \\ \hline &9&45&225&\color{blue}{1125}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ 5625 } = \color{orangered}{ 5624 } $
$$ \begin{array}{c|rrrrrr}5&9&0&0&0&\color{orangered}{ -1 }&24\\& & 45& 225& 1125& \color{orangered}{5625} & \\ \hline &9&45&225&1125&\color{orangered}{5624}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 5624 } = \color{blue}{ 28120 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{5}&9&0&0&0&-1&24\\& & 45& 225& 1125& 5625& \color{blue}{28120} \\ \hline &9&45&225&1125&\color{blue}{5624}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 24 } + \color{orangered}{ 28120 } = \color{orangered}{ 28144 } $
$$ \begin{array}{c|rrrrrr}5&9&0&0&0&-1&\color{orangered}{ 24 }\\& & 45& 225& 1125& 5625& \color{orangered}{28120} \\ \hline &\color{blue}{9}&\color{blue}{45}&\color{blue}{225}&\color{blue}{1125}&\color{blue}{5624}&\color{orangered}{28144} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 9x^{4}+45x^{3}+225x^{2}+1125x+5624 } $ with a remainder of $ \color{red}{ 28144 } $.