Divide using synthetic division $$ ( -3x^{3}+2x-1 ) \div ( x+9 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-9&-3&0&2&-1\\& & 27& -243& \color{black}{2169} \\ \hline &\color{blue}{-3}&\color{blue}{27}&\color{blue}{-241}&\color{orangered}{2168} \end{array} $$The solution is:
$$ \frac{ -3x^{3}+2x-1 }{ x+9 } = \color{blue}{-3x^{2}+27x-241} ~+~ \frac{ \color{red}{ 2168 } }{ x+9 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-9}&-3&0&2&-1\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-9&\color{orangered}{ -3 }&0&2&-1\\& & & & \\ \hline &\color{orangered}{-3}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -9 } \cdot \color{blue}{ \left( -3 \right) } = \color{blue}{ 27 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-9}&-3&0&2&-1\\& & \color{blue}{27} & & \\ \hline &\color{blue}{-3}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 27 } = \color{orangered}{ 27 } $
$$ \begin{array}{c|rrrr}-9&-3&\color{orangered}{ 0 }&2&-1\\& & \color{orangered}{27} & & \\ \hline &-3&\color{orangered}{27}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -9 } \cdot \color{blue}{ 27 } = \color{blue}{ -243 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-9}&-3&0&2&-1\\& & 27& \color{blue}{-243} & \\ \hline &-3&\color{blue}{27}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ \left( -243 \right) } = \color{orangered}{ -241 } $
$$ \begin{array}{c|rrrr}-9&-3&0&\color{orangered}{ 2 }&-1\\& & 27& \color{orangered}{-243} & \\ \hline &-3&27&\color{orangered}{-241}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -9 } \cdot \color{blue}{ \left( -241 \right) } = \color{blue}{ 2169 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-9}&-3&0&2&-1\\& & 27& -243& \color{blue}{2169} \\ \hline &-3&27&\color{blue}{-241}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ 2169 } = \color{orangered}{ 2168 } $
$$ \begin{array}{c|rrrr}-9&-3&0&2&\color{orangered}{ -1 }\\& & 27& -243& \color{orangered}{2169} \\ \hline &\color{blue}{-3}&\color{blue}{27}&\color{blue}{-241}&\color{orangered}{2168} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -3x^{2}+27x-241 } $ with a remainder of $ \color{red}{ 2168 } $.