Divide using synthetic division $$ ( x^{3}+x^{2}+x+1 ) \div ( x+9 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-9&1&1&1&1\\& & -9& 72& \color{black}{-657} \\ \hline &\color{blue}{1}&\color{blue}{-8}&\color{blue}{73}&\color{orangered}{-656} \end{array} $$The solution is:
$$ \frac{ x^{3}+x^{2}+x+1 }{ x+9 } = \color{blue}{x^{2}-8x+73} \color{red}{~-~} \frac{ \color{red}{ 656 } }{ x+9 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-9}&1&1&1&1\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-9&\color{orangered}{ 1 }&1&1&1\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-9}&1&1&1&1\\& & \color{blue}{-9} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ \left( -9 \right) } = \color{orangered}{ -8 } $
$$ \begin{array}{c|rrrr}-9&1&\color{orangered}{ 1 }&1&1\\& & \color{orangered}{-9} & & \\ \hline &1&\color{orangered}{-8}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -9 } \cdot \color{blue}{ \left( -8 \right) } = \color{blue}{ 72 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-9}&1&1&1&1\\& & -9& \color{blue}{72} & \\ \hline &1&\color{blue}{-8}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ 72 } = \color{orangered}{ 73 } $
$$ \begin{array}{c|rrrr}-9&1&1&\color{orangered}{ 1 }&1\\& & -9& \color{orangered}{72} & \\ \hline &1&-8&\color{orangered}{73}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -9 } \cdot \color{blue}{ 73 } = \color{blue}{ -657 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-9}&1&1&1&1\\& & -9& 72& \color{blue}{-657} \\ \hline &1&-8&\color{blue}{73}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ \left( -657 \right) } = \color{orangered}{ -656 } $
$$ \begin{array}{c|rrrr}-9&1&1&1&\color{orangered}{ 1 }\\& & -9& 72& \color{orangered}{-657} \\ \hline &\color{blue}{1}&\color{blue}{-8}&\color{blue}{73}&\color{orangered}{-656} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{2}-8x+73 } $ with a remainder of $ \color{red}{ -656 } $.