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

The synthetic division table is:

$$ \begin{array}{c|rrrr}-5&-1&-5&0&3\\& & 5& 0& \color{black}{0} \\ \hline &\color{blue}{-1}&\color{blue}{0}&\color{blue}{0}&\color{orangered}{3} \end{array} $$

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

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 + 5 = 0 $ ( $ x = \color{blue}{ -5 } $ ) at the left.

$$ \begin{array}{c|rrrr}\color{blue}{-5}&-1&-5&0&3\\& & & & \\ \hline &&&& \end{array} $$

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

$$ \begin{array}{c|rrrr}-5&\color{orangered}{ -1 }&-5&0&3\\& & & & \\ \hline &\color{orangered}{-1}&&& \end{array} $$

Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ 5 } $.

$$ \begin{array}{c|rrrr}\color{blue}{-5}&-1&-5&0&3\\& & \color{blue}{5} & & \\ \hline &\color{blue}{-1}&&& \end{array} $$

Step 3 : Add down last column: $ \color{orangered}{ -5 } + \color{orangered}{ 5 } = \color{orangered}{ 0 } $

$$ \begin{array}{c|rrrr}-5&-1&\color{orangered}{ -5 }&0&3\\& & \color{orangered}{5} & & \\ \hline &-1&\color{orangered}{0}&& \end{array} $$

Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.

$$ \begin{array}{c|rrrr}\color{blue}{-5}&-1&-5&0&3\\& & 5& \color{blue}{0} & \\ \hline &-1&\color{blue}{0}&& \end{array} $$

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

$$ \begin{array}{c|rrrr}-5&-1&-5&\color{orangered}{ 0 }&3\\& & 5& \color{orangered}{0} & \\ \hline &-1&0&\color{orangered}{0}& \end{array} $$

Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.

$$ \begin{array}{c|rrrr}\color{blue}{-5}&-1&-5&0&3\\& & 5& 0& \color{blue}{0} \\ \hline &-1&0&\color{blue}{0}& \end{array} $$

Step 7 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ 0 } = \color{orangered}{ 3 } $

$$ \begin{array}{c|rrrr}-5&-1&-5&0&\color{orangered}{ 3 }\\& & 5& 0& \color{orangered}{0} \\ \hline &\color{blue}{-1}&\color{blue}{0}&\color{blue}{0}&\color{orangered}{3} \end{array} $$

Remainder is the last entry in the bottom row $ \left(\color{red}{ 3 }\right) $.

Synthetic Division Calculator