Divide using synthetic division $$ ( 3x^{5}-3x^{4}+2x^{3}-x+9 ) \div ( x+2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}-2&3&-3&2&0&-1&9\\& & -6& 18& -40& 80& \color{black}{-158} \\ \hline &\color{blue}{3}&\color{blue}{-9}&\color{blue}{20}&\color{blue}{-40}&\color{blue}{79}&\color{orangered}{-149} \end{array} $$The solution is:
$$ \frac{ 3x^{5}-3x^{4}+2x^{3}-x+9 }{ x+2 } = \color{blue}{3x^{4}-9x^{3}+20x^{2}-40x+79} \color{red}{~-~} \frac{ \color{red}{ 149 } }{ x+2 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 2 = 0 $ ( $ x = \color{blue}{ -2 } $ ) at the left.
$$ \begin{array}{c|rrrrrr}\color{blue}{-2}&3&-3&2&0&-1&9\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}-2&\color{orangered}{ 3 }&-3&2&0&-1&9\\& & & & & & \\ \hline &\color{orangered}{3}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 3 } = \color{blue}{ -6 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-2}&3&-3&2&0&-1&9\\& & \color{blue}{-6} & & & & \\ \hline &\color{blue}{3}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ \left( -6 \right) } = \color{orangered}{ -9 } $
$$ \begin{array}{c|rrrrrr}-2&3&\color{orangered}{ -3 }&2&0&-1&9\\& & \color{orangered}{-6} & & & & \\ \hline &3&\color{orangered}{-9}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -9 \right) } = \color{blue}{ 18 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-2}&3&-3&2&0&-1&9\\& & -6& \color{blue}{18} & & & \\ \hline &3&\color{blue}{-9}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ 18 } = \color{orangered}{ 20 } $
$$ \begin{array}{c|rrrrrr}-2&3&-3&\color{orangered}{ 2 }&0&-1&9\\& & -6& \color{orangered}{18} & & & \\ \hline &3&-9&\color{orangered}{20}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 20 } = \color{blue}{ -40 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-2}&3&-3&2&0&-1&9\\& & -6& 18& \color{blue}{-40} & & \\ \hline &3&-9&\color{blue}{20}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -40 \right) } = \color{orangered}{ -40 } $
$$ \begin{array}{c|rrrrrr}-2&3&-3&2&\color{orangered}{ 0 }&-1&9\\& & -6& 18& \color{orangered}{-40} & & \\ \hline &3&-9&20&\color{orangered}{-40}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -40 \right) } = \color{blue}{ 80 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-2}&3&-3&2&0&-1&9\\& & -6& 18& -40& \color{blue}{80} & \\ \hline &3&-9&20&\color{blue}{-40}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ 80 } = \color{orangered}{ 79 } $
$$ \begin{array}{c|rrrrrr}-2&3&-3&2&0&\color{orangered}{ -1 }&9\\& & -6& 18& -40& \color{orangered}{80} & \\ \hline &3&-9&20&-40&\color{orangered}{79}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 79 } = \color{blue}{ -158 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-2}&3&-3&2&0&-1&9\\& & -6& 18& -40& 80& \color{blue}{-158} \\ \hline &3&-9&20&-40&\color{blue}{79}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 9 } + \color{orangered}{ \left( -158 \right) } = \color{orangered}{ -149 } $
$$ \begin{array}{c|rrrrrr}-2&3&-3&2&0&-1&\color{orangered}{ 9 }\\& & -6& 18& -40& 80& \color{orangered}{-158} \\ \hline &\color{blue}{3}&\color{blue}{-9}&\color{blue}{20}&\color{blue}{-40}&\color{blue}{79}&\color{orangered}{-149} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{4}-9x^{3}+20x^{2}-40x+79 } $ with a remainder of $ \color{red}{ -149 } $.