Divide using synthetic division $$ ( 9x^{4}-9x^{3}-58x^{2}+5x+24 ) \div ( x-24 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}24&9&-9&-58&5&24\\& & 216& 4968& 117840& \color{black}{2828280} \\ \hline &\color{blue}{9}&\color{blue}{207}&\color{blue}{4910}&\color{blue}{117845}&\color{orangered}{2828304} \end{array} $$The solution is:
$$ \frac{ 9x^{4}-9x^{3}-58x^{2}+5x+24 }{ x-24 } = \color{blue}{9x^{3}+207x^{2}+4910x+117845} ~+~ \frac{ \color{red}{ 2828304 } }{ x-24 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -24 = 0 $ ( $ x = \color{blue}{ 24 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{24}&9&-9&-58&5&24\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}24&\color{orangered}{ 9 }&-9&-58&5&24\\& & & & & \\ \hline &\color{orangered}{9}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 24 } \cdot \color{blue}{ 9 } = \color{blue}{ 216 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{24}&9&-9&-58&5&24\\& & \color{blue}{216} & & & \\ \hline &\color{blue}{9}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -9 } + \color{orangered}{ 216 } = \color{orangered}{ 207 } $
$$ \begin{array}{c|rrrrr}24&9&\color{orangered}{ -9 }&-58&5&24\\& & \color{orangered}{216} & & & \\ \hline &9&\color{orangered}{207}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 24 } \cdot \color{blue}{ 207 } = \color{blue}{ 4968 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{24}&9&-9&-58&5&24\\& & 216& \color{blue}{4968} & & \\ \hline &9&\color{blue}{207}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -58 } + \color{orangered}{ 4968 } = \color{orangered}{ 4910 } $
$$ \begin{array}{c|rrrrr}24&9&-9&\color{orangered}{ -58 }&5&24\\& & 216& \color{orangered}{4968} & & \\ \hline &9&207&\color{orangered}{4910}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 24 } \cdot \color{blue}{ 4910 } = \color{blue}{ 117840 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{24}&9&-9&-58&5&24\\& & 216& 4968& \color{blue}{117840} & \\ \hline &9&207&\color{blue}{4910}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ 117840 } = \color{orangered}{ 117845 } $
$$ \begin{array}{c|rrrrr}24&9&-9&-58&\color{orangered}{ 5 }&24\\& & 216& 4968& \color{orangered}{117840} & \\ \hline &9&207&4910&\color{orangered}{117845}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 24 } \cdot \color{blue}{ 117845 } = \color{blue}{ 2828280 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{24}&9&-9&-58&5&24\\& & 216& 4968& 117840& \color{blue}{2828280} \\ \hline &9&207&4910&\color{blue}{117845}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 24 } + \color{orangered}{ 2828280 } = \color{orangered}{ 2828304 } $
$$ \begin{array}{c|rrrrr}24&9&-9&-58&5&\color{orangered}{ 24 }\\& & 216& 4968& 117840& \color{orangered}{2828280} \\ \hline &\color{blue}{9}&\color{blue}{207}&\color{blue}{4910}&\color{blue}{117845}&\color{orangered}{2828304} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 9x^{3}+207x^{2}+4910x+117845 } $ with a remainder of $ \color{red}{ 2828304 } $.