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

The synthetic division table is:

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

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

We can find remainder using synthetic division method.

Step 1 : Write down the coefficients of the dividend into division table.Put the zero at the left.

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

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

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

Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ 0 } $.

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

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

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

Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ \left( -3 \right) } = \color{blue}{ 0 } $.

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

Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 0 } = \color{orangered}{ 0 } $

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

Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.

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

Step 7 : Add down last column: $ \color{orangered}{ -4 } + \color{orangered}{ 0 } = \color{orangered}{ -4 } $

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

Remainder is the last entry in the bottom row $ \left(\color{red}{ -4 }\right) $.

Synthetic Division Calculator