Find remainder when $ 3x^{5}-2x^{3}+3x-2 $ is divided by $ x+1 $.

The synthetic division table is:

$$ \begin{array}{c|rrrrrr}-1&3&0&-2&0&3&-2\\& & -3& 3& -1& 1& \color{black}{-4} \\ \hline &\color{blue}{3}&\color{blue}{-3}&\color{blue}{1}&\color{blue}{-1}&\color{blue}{4}&\color{orangered}{-6} \end{array} $$

The remainder when $ 3x^{5}-2x^{3}+3x-2 $ is divided by $ x+1 $ is $ \, \color{red}{ -6 } $.

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.

$$ \begin{array}{c|rrrrrr}\color{blue}{-1}&3&0&-2&0&3&-2\\& & & & & & \\ \hline &&&&&& \end{array} $$

Step 1 : Bring down the leading coefficient to the bottom row.

$$ \begin{array}{c|rrrrrr}-1&\color{orangered}{ 3 }&0&-2&0&3&-2\\& & & & & & \\ \hline &\color{orangered}{3}&&&&& \end{array} $$

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 } $.

$$ \begin{array}{c|rrrrrr}\color{blue}{-1}&3&0&-2&0&3&-2\\& & \color{blue}{-3} & & & & \\ \hline &\color{blue}{3}&&&&& \end{array} $$

Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -3 \right) } = \color{orangered}{ -3 } $

$$ \begin{array}{c|rrrrrr}-1&3&\color{orangered}{ 0 }&-2&0&3&-2\\& & \color{orangered}{-3} & & & & \\ \hline &3&\color{orangered}{-3}&&&& \end{array} $$

Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -3 \right) } = \color{blue}{ 3 } $.

$$ \begin{array}{c|rrrrrr}\color{blue}{-1}&3&0&-2&0&3&-2\\& & -3& \color{blue}{3} & & & \\ \hline &3&\color{blue}{-3}&&&& \end{array} $$

Step 5 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ 3 } = \color{orangered}{ 1 } $

$$ \begin{array}{c|rrrrrr}-1&3&0&\color{orangered}{ -2 }&0&3&-2\\& & -3& \color{orangered}{3} & & & \\ \hline &3&-3&\color{orangered}{1}&&& \end{array} $$

Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 1 } = \color{blue}{ -1 } $.

$$ \begin{array}{c|rrrrrr}\color{blue}{-1}&3&0&-2&0&3&-2\\& & -3& 3& \color{blue}{-1} & & \\ \hline &3&-3&\color{blue}{1}&&& \end{array} $$

Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -1 \right) } = \color{orangered}{ -1 } $

$$ \begin{array}{c|rrrrrr}-1&3&0&-2&\color{orangered}{ 0 }&3&-2\\& & -3& 3& \color{orangered}{-1} & & \\ \hline &3&-3&1&\color{orangered}{-1}&& \end{array} $$

Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ 1 } $.

$$ \begin{array}{c|rrrrrr}\color{blue}{-1}&3&0&-2&0&3&-2\\& & -3& 3& -1& \color{blue}{1} & \\ \hline &3&-3&1&\color{blue}{-1}&& \end{array} $$

Step 9 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ 1 } = \color{orangered}{ 4 } $

$$ \begin{array}{c|rrrrrr}-1&3&0&-2&0&\color{orangered}{ 3 }&-2\\& & -3& 3& -1& \color{orangered}{1} & \\ \hline &3&-3&1&-1&\color{orangered}{4}& \end{array} $$

Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 4 } = \color{blue}{ -4 } $.

$$ \begin{array}{c|rrrrrr}\color{blue}{-1}&3&0&-2&0&3&-2\\& & -3& 3& -1& 1& \color{blue}{-4} \\ \hline &3&-3&1&-1&\color{blue}{4}& \end{array} $$

Step 11 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ \left( -4 \right) } = \color{orangered}{ -6 } $

$$ \begin{array}{c|rrrrrr}-1&3&0&-2&0&3&\color{orangered}{ -2 }\\& & -3& 3& -1& 1& \color{orangered}{-4} \\ \hline &\color{blue}{3}&\color{blue}{-3}&\color{blue}{1}&\color{blue}{-1}&\color{blue}{4}&\color{orangered}{-6} \end{array} $$

Remainder is the last entry in the bottom row $ \left(\color{red}{ -6 }\right) $.

Synthetic Division Calculator