Find remainder when $ 2x^{3}+x^{2}-2x+4 $ is divided by $ x+1 $.

The synthetic division table is:

$$ \begin{array}{c|rrrr}-1&2&1&-2&4\\& & -2& 1& \color{black}{1} \\ \hline &\color{blue}{2}&\color{blue}{-1}&\color{blue}{-1}&\color{orangered}{5} \end{array} $$

The remainder when $ 2x^{3}+x^{2}-2x+4 $ is divided by $ x+1 $ is $ \, \color{red}{ 5 } $.

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.

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

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

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

Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 2 } = \color{blue}{ -2 } $.

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

Step 3 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ \left( -2 \right) } = \color{orangered}{ -1 } $

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

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

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

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

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

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

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

Step 7 : Add down last column: $ \color{orangered}{ 4 } + \color{orangered}{ 1 } = \color{orangered}{ 5 } $

$$ \begin{array}{c|rrrr}-1&2&1&-2&\color{orangered}{ 4 }\\& & -2& 1& \color{orangered}{1} \\ \hline &\color{blue}{2}&\color{blue}{-1}&\color{blue}{-1}&\color{orangered}{5} \end{array} $$

Remainder is the last entry in the bottom row $ \left(\color{red}{ 5 }\right) $.

Synthetic Division Calculator