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