Find remainder when $ 2x^{5}-4x^{4}+3x^{3}-x+9 $ is divided by $ x+3 $.
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}-3&2&-4&3&0&-1&9\\& & -6& 30& -99& 297& \color{black}{-888} \\ \hline &\color{blue}{2}&\color{blue}{-10}&\color{blue}{33}&\color{blue}{-99}&\color{blue}{296}&\color{orangered}{-879} \end{array} $$The remainder when $ 2x^{5}-4x^{4}+3x^{3}-x+9 $ is divided by $ x+3 $ is $ \, \color{red}{ -879 } $.
We can find remainder using synthetic division method.
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 3 = 0 $ ( $ x = \color{blue}{ -3 } $ ) at the left.
$$ \begin{array}{c|rrrrrr}\color{blue}{-3}&2&-4&3&0&-1&9\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}-3&\color{orangered}{ 2 }&-4&3&0&-1&9\\& & & & & & \\ \hline &\color{orangered}{2}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 2 } = \color{blue}{ -6 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-3}&2&-4&3&0&-1&9\\& & \color{blue}{-6} & & & & \\ \hline &\color{blue}{2}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -4 } + \color{orangered}{ \left( -6 \right) } = \color{orangered}{ -10 } $
$$ \begin{array}{c|rrrrrr}-3&2&\color{orangered}{ -4 }&3&0&-1&9\\& & \color{orangered}{-6} & & & & \\ \hline &2&\color{orangered}{-10}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -10 \right) } = \color{blue}{ 30 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-3}&2&-4&3&0&-1&9\\& & -6& \color{blue}{30} & & & \\ \hline &2&\color{blue}{-10}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ 30 } = \color{orangered}{ 33 } $
$$ \begin{array}{c|rrrrrr}-3&2&-4&\color{orangered}{ 3 }&0&-1&9\\& & -6& \color{orangered}{30} & & & \\ \hline &2&-10&\color{orangered}{33}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 33 } = \color{blue}{ -99 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-3}&2&-4&3&0&-1&9\\& & -6& 30& \color{blue}{-99} & & \\ \hline &2&-10&\color{blue}{33}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -99 \right) } = \color{orangered}{ -99 } $
$$ \begin{array}{c|rrrrrr}-3&2&-4&3&\color{orangered}{ 0 }&-1&9\\& & -6& 30& \color{orangered}{-99} & & \\ \hline &2&-10&33&\color{orangered}{-99}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -99 \right) } = \color{blue}{ 297 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-3}&2&-4&3&0&-1&9\\& & -6& 30& -99& \color{blue}{297} & \\ \hline &2&-10&33&\color{blue}{-99}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ 297 } = \color{orangered}{ 296 } $
$$ \begin{array}{c|rrrrrr}-3&2&-4&3&0&\color{orangered}{ -1 }&9\\& & -6& 30& -99& \color{orangered}{297} & \\ \hline &2&-10&33&-99&\color{orangered}{296}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 296 } = \color{blue}{ -888 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-3}&2&-4&3&0&-1&9\\& & -6& 30& -99& 297& \color{blue}{-888} \\ \hline &2&-10&33&-99&\color{blue}{296}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 9 } + \color{orangered}{ \left( -888 \right) } = \color{orangered}{ -879 } $
$$ \begin{array}{c|rrrrrr}-3&2&-4&3&0&-1&\color{orangered}{ 9 }\\& & -6& 30& -99& 297& \color{orangered}{-888} \\ \hline &\color{blue}{2}&\color{blue}{-10}&\color{blue}{33}&\color{blue}{-99}&\color{blue}{296}&\color{orangered}{-879} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ -879 }\right) $.