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

The synthetic division table is:

$$ \begin{array}{c|rrr}-1&-1&0&-1\\& & 1& \color{black}{-1} \\ \hline &\color{blue}{-1}&\color{blue}{1}&\color{orangered}{-2} \end{array} $$

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

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.

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

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

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

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

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

Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 1 } = \color{orangered}{ 1 } $

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

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

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

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

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

Remainder is the last entry in the bottom row $ \left(\color{red}{ -2 }\right) $.

Synthetic Division Calculator