Divide using synthetic division $$ ( 9x^{4}-9x^{3}-58x^{2}+5x+24 ) \div ( x-12 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}12&9&-9&-58&5&24\\& & 108& 1188& 13560& \color{black}{162780} \\ \hline &\color{blue}{9}&\color{blue}{99}&\color{blue}{1130}&\color{blue}{13565}&\color{orangered}{162804} \end{array} $$The solution is:
$$ \frac{ 9x^{4}-9x^{3}-58x^{2}+5x+24 }{ x-12 } = \color{blue}{9x^{3}+99x^{2}+1130x+13565} ~+~ \frac{ \color{red}{ 162804 } }{ x-12 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -12 = 0 $ ( $ x = \color{blue}{ 12 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{12}&9&-9&-58&5&24\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}12&\color{orangered}{ 9 }&-9&-58&5&24\\& & & & & \\ \hline &\color{orangered}{9}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 12 } \cdot \color{blue}{ 9 } = \color{blue}{ 108 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{12}&9&-9&-58&5&24\\& & \color{blue}{108} & & & \\ \hline &\color{blue}{9}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -9 } + \color{orangered}{ 108 } = \color{orangered}{ 99 } $
$$ \begin{array}{c|rrrrr}12&9&\color{orangered}{ -9 }&-58&5&24\\& & \color{orangered}{108} & & & \\ \hline &9&\color{orangered}{99}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 12 } \cdot \color{blue}{ 99 } = \color{blue}{ 1188 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{12}&9&-9&-58&5&24\\& & 108& \color{blue}{1188} & & \\ \hline &9&\color{blue}{99}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -58 } + \color{orangered}{ 1188 } = \color{orangered}{ 1130 } $
$$ \begin{array}{c|rrrrr}12&9&-9&\color{orangered}{ -58 }&5&24\\& & 108& \color{orangered}{1188} & & \\ \hline &9&99&\color{orangered}{1130}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 12 } \cdot \color{blue}{ 1130 } = \color{blue}{ 13560 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{12}&9&-9&-58&5&24\\& & 108& 1188& \color{blue}{13560} & \\ \hline &9&99&\color{blue}{1130}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ 13560 } = \color{orangered}{ 13565 } $
$$ \begin{array}{c|rrrrr}12&9&-9&-58&\color{orangered}{ 5 }&24\\& & 108& 1188& \color{orangered}{13560} & \\ \hline &9&99&1130&\color{orangered}{13565}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 12 } \cdot \color{blue}{ 13565 } = \color{blue}{ 162780 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{12}&9&-9&-58&5&24\\& & 108& 1188& 13560& \color{blue}{162780} \\ \hline &9&99&1130&\color{blue}{13565}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 24 } + \color{orangered}{ 162780 } = \color{orangered}{ 162804 } $
$$ \begin{array}{c|rrrrr}12&9&-9&-58&5&\color{orangered}{ 24 }\\& & 108& 1188& 13560& \color{orangered}{162780} \\ \hline &\color{blue}{9}&\color{blue}{99}&\color{blue}{1130}&\color{blue}{13565}&\color{orangered}{162804} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 9x^{3}+99x^{2}+1130x+13565 } $ with a remainder of $ \color{red}{ 162804 } $.