Find remainder when $ 3x^{3}-10x^{2}+3x-4 $ is divided by $ x+1 $.
The synthetic division table is:
$$ \begin{array}{c|rrrr}-1&3&-10&3&-4\\& & -3& 13& \color{black}{-16} \\ \hline &\color{blue}{3}&\color{blue}{-13}&\color{blue}{16}&\color{orangered}{-20} \end{array} $$The remainder when $ 3x^{3}-10x^{2}+3x-4 $ is divided by $ x+1 $ is $ \, \color{red}{ -20 } $.
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 + 1 = 0 $ ( $ x = \color{blue}{ -1 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&3&-10&3&-4\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-1&\color{orangered}{ 3 }&-10&3&-4\\& & & & \\ \hline &\color{orangered}{3}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 3 } = \color{blue}{ -3 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&3&-10&3&-4\\& & \color{blue}{-3} & & \\ \hline &\color{blue}{3}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -10 } + \color{orangered}{ \left( -3 \right) } = \color{orangered}{ -13 } $
$$ \begin{array}{c|rrrr}-1&3&\color{orangered}{ -10 }&3&-4\\& & \color{orangered}{-3} & & \\ \hline &3&\color{orangered}{-13}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -13 \right) } = \color{blue}{ 13 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&3&-10&3&-4\\& & -3& \color{blue}{13} & \\ \hline &3&\color{blue}{-13}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ 13 } = \color{orangered}{ 16 } $
$$ \begin{array}{c|rrrr}-1&3&-10&\color{orangered}{ 3 }&-4\\& & -3& \color{orangered}{13} & \\ \hline &3&-13&\color{orangered}{16}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 16 } = \color{blue}{ -16 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&3&-10&3&-4\\& & -3& 13& \color{blue}{-16} \\ \hline &3&-13&\color{blue}{16}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -4 } + \color{orangered}{ \left( -16 \right) } = \color{orangered}{ -20 } $
$$ \begin{array}{c|rrrr}-1&3&-10&3&\color{orangered}{ -4 }\\& & -3& 13& \color{orangered}{-16} \\ \hline &\color{blue}{3}&\color{blue}{-13}&\color{blue}{16}&\color{orangered}{-20} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ -20 }\right) $.