Divide using synthetic division $$ ( 9x^{4}-9x^{3}-58x^{2}+5x+24 ) \div ( x+3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-3&9&-9&-58&5&24\\& & -27& 108& -150& \color{black}{435} \\ \hline &\color{blue}{9}&\color{blue}{-36}&\color{blue}{50}&\color{blue}{-145}&\color{orangered}{459} \end{array} $$The solution is:
$$ \frac{ 9x^{4}-9x^{3}-58x^{2}+5x+24 }{ x+3 } = \color{blue}{9x^{3}-36x^{2}+50x-145} ~+~ \frac{ \color{red}{ 459 } }{ x+3 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 3 = 0 $ ( $ x = \color{blue}{ -3 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&9&-9&-58&5&24\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-3&\color{orangered}{ 9 }&-9&-58&5&24\\& & & & & \\ \hline &\color{orangered}{9}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 9 } = \color{blue}{ -27 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&9&-9&-58&5&24\\& & \color{blue}{-27} & & & \\ \hline &\color{blue}{9}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -9 } + \color{orangered}{ \left( -27 \right) } = \color{orangered}{ -36 } $
$$ \begin{array}{c|rrrrr}-3&9&\color{orangered}{ -9 }&-58&5&24\\& & \color{orangered}{-27} & & & \\ \hline &9&\color{orangered}{-36}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -36 \right) } = \color{blue}{ 108 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&9&-9&-58&5&24\\& & -27& \color{blue}{108} & & \\ \hline &9&\color{blue}{-36}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -58 } + \color{orangered}{ 108 } = \color{orangered}{ 50 } $
$$ \begin{array}{c|rrrrr}-3&9&-9&\color{orangered}{ -58 }&5&24\\& & -27& \color{orangered}{108} & & \\ \hline &9&-36&\color{orangered}{50}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 50 } = \color{blue}{ -150 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&9&-9&-58&5&24\\& & -27& 108& \color{blue}{-150} & \\ \hline &9&-36&\color{blue}{50}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ \left( -150 \right) } = \color{orangered}{ -145 } $
$$ \begin{array}{c|rrrrr}-3&9&-9&-58&\color{orangered}{ 5 }&24\\& & -27& 108& \color{orangered}{-150} & \\ \hline &9&-36&50&\color{orangered}{-145}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -145 \right) } = \color{blue}{ 435 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&9&-9&-58&5&24\\& & -27& 108& -150& \color{blue}{435} \\ \hline &9&-36&50&\color{blue}{-145}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 24 } + \color{orangered}{ 435 } = \color{orangered}{ 459 } $
$$ \begin{array}{c|rrrrr}-3&9&-9&-58&5&\color{orangered}{ 24 }\\& & -27& 108& -150& \color{orangered}{435} \\ \hline &\color{blue}{9}&\color{blue}{-36}&\color{blue}{50}&\color{blue}{-145}&\color{orangered}{459} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 9x^{3}-36x^{2}+50x-145 } $ with a remainder of $ \color{red}{ 459 } $.