Divide using synthetic division $$ ( 9x^{4}-9x^{3}-58x^{2}+5x+24 ) \div ( x-4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}4&9&-9&-58&5&24\\& & 36& 108& 200& \color{black}{820} \\ \hline &\color{blue}{9}&\color{blue}{27}&\color{blue}{50}&\color{blue}{205}&\color{orangered}{844} \end{array} $$The solution is:
$$ \frac{ 9x^{4}-9x^{3}-58x^{2}+5x+24 }{ x-4 } = \color{blue}{9x^{3}+27x^{2}+50x+205} ~+~ \frac{ \color{red}{ 844 } }{ x-4 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -4 = 0 $ ( $ x = \color{blue}{ 4 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&9&-9&-58&5&24\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}4&\color{orangered}{ 9 }&-9&-58&5&24\\& & & & & \\ \hline &\color{orangered}{9}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 9 } = \color{blue}{ 36 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&9&-9&-58&5&24\\& & \color{blue}{36} & & & \\ \hline &\color{blue}{9}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -9 } + \color{orangered}{ 36 } = \color{orangered}{ 27 } $
$$ \begin{array}{c|rrrrr}4&9&\color{orangered}{ -9 }&-58&5&24\\& & \color{orangered}{36} & & & \\ \hline &9&\color{orangered}{27}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 27 } = \color{blue}{ 108 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&9&-9&-58&5&24\\& & 36& \color{blue}{108} & & \\ \hline &9&\color{blue}{27}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -58 } + \color{orangered}{ 108 } = \color{orangered}{ 50 } $
$$ \begin{array}{c|rrrrr}4&9&-9&\color{orangered}{ -58 }&5&24\\& & 36& \color{orangered}{108} & & \\ \hline &9&27&\color{orangered}{50}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 50 } = \color{blue}{ 200 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&9&-9&-58&5&24\\& & 36& 108& \color{blue}{200} & \\ \hline &9&27&\color{blue}{50}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ 200 } = \color{orangered}{ 205 } $
$$ \begin{array}{c|rrrrr}4&9&-9&-58&\color{orangered}{ 5 }&24\\& & 36& 108& \color{orangered}{200} & \\ \hline &9&27&50&\color{orangered}{205}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 205 } = \color{blue}{ 820 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&9&-9&-58&5&24\\& & 36& 108& 200& \color{blue}{820} \\ \hline &9&27&50&\color{blue}{205}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 24 } + \color{orangered}{ 820 } = \color{orangered}{ 844 } $
$$ \begin{array}{c|rrrrr}4&9&-9&-58&5&\color{orangered}{ 24 }\\& & 36& 108& 200& \color{orangered}{820} \\ \hline &\color{blue}{9}&\color{blue}{27}&\color{blue}{50}&\color{blue}{205}&\color{orangered}{844} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 9x^{3}+27x^{2}+50x+205 } $ with a remainder of $ \color{red}{ 844 } $.