Divide using synthetic division $$ ( x^{4}-x^{2}+9 ) \div ( x+9 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-9&1&0&-1&0&9\\& & -9& 81& -720& \color{black}{6480} \\ \hline &\color{blue}{1}&\color{blue}{-9}&\color{blue}{80}&\color{blue}{-720}&\color{orangered}{6489} \end{array} $$The solution is:
$$ \frac{ x^{4}-x^{2}+9 }{ x+9 } = \color{blue}{x^{3}-9x^{2}+80x-720} ~+~ \frac{ \color{red}{ 6489 } }{ x+9 } $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{-9}&1&0&-1&0&9\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-9&\color{orangered}{ 1 }&0&-1&0&9\\& & & & & \\ \hline &\color{orangered}{1}&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-9}&1&0&-1&0&9\\& & \color{blue}{-9} & & & \\ \hline &\color{blue}{1}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -9 \right) } = \color{orangered}{ -9 } $
$$ \begin{array}{c|rrrrr}-9&1&\color{orangered}{ 0 }&-1&0&9\\& & \color{orangered}{-9} & & & \\ \hline &1&\color{orangered}{-9}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -9 } \cdot \color{blue}{ \left( -9 \right) } = \color{blue}{ 81 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-9}&1&0&-1&0&9\\& & -9& \color{blue}{81} & & \\ \hline &1&\color{blue}{-9}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ 81 } = \color{orangered}{ 80 } $
$$ \begin{array}{c|rrrrr}-9&1&0&\color{orangered}{ -1 }&0&9\\& & -9& \color{orangered}{81} & & \\ \hline &1&-9&\color{orangered}{80}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -9 } \cdot \color{blue}{ 80 } = \color{blue}{ -720 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-9}&1&0&-1&0&9\\& & -9& 81& \color{blue}{-720} & \\ \hline &1&-9&\color{blue}{80}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -720 \right) } = \color{orangered}{ -720 } $
$$ \begin{array}{c|rrrrr}-9&1&0&-1&\color{orangered}{ 0 }&9\\& & -9& 81& \color{orangered}{-720} & \\ \hline &1&-9&80&\color{orangered}{-720}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -9 } \cdot \color{blue}{ \left( -720 \right) } = \color{blue}{ 6480 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-9}&1&0&-1&0&9\\& & -9& 81& -720& \color{blue}{6480} \\ \hline &1&-9&80&\color{blue}{-720}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 9 } + \color{orangered}{ 6480 } = \color{orangered}{ 6489 } $
$$ \begin{array}{c|rrrrr}-9&1&0&-1&0&\color{orangered}{ 9 }\\& & -9& 81& -720& \color{orangered}{6480} \\ \hline &\color{blue}{1}&\color{blue}{-9}&\color{blue}{80}&\color{blue}{-720}&\color{orangered}{6489} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{3}-9x^{2}+80x-720 } $ with a remainder of $ \color{red}{ 6489 } $.