Find remainder when $ x^{4}+5x^{3}-50x^{2}-x $ is divided by $ x+10 $.

The synthetic division table is:

$$ \begin{array}{c|rrrrr}-10&1&5&-50&-1&0\\& & -10& 50& 0& \color{black}{10} \\ \hline &\color{blue}{1}&\color{blue}{-5}&\color{blue}{0}&\color{blue}{-1}&\color{orangered}{10} \end{array} $$

The remainder when $ x^{4}+5x^{3}-50x^{2}-x $ is divided by $ x+10 $ is $ \, \color{red}{ 10 } $.

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 + 10 = 0 $ ( $ x = \color{blue}{ -10 } $ ) at the left.

$$ \begin{array}{c|rrrrr}\color{blue}{-10}&1&5&-50&-1&0\\& & & & & \\ \hline &&&&& \end{array} $$

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

$$ \begin{array}{c|rrrrr}-10&\color{orangered}{ 1 }&5&-50&-1&0\\& & & & & \\ \hline &\color{orangered}{1}&&&& \end{array} $$

Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -10 } \cdot \color{blue}{ 1 } = \color{blue}{ -10 } $.

$$ \begin{array}{c|rrrrr}\color{blue}{-10}&1&5&-50&-1&0\\& & \color{blue}{-10} & & & \\ \hline &\color{blue}{1}&&&& \end{array} $$

Step 3 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ \left( -10 \right) } = \color{orangered}{ -5 } $

$$ \begin{array}{c|rrrrr}-10&1&\color{orangered}{ 5 }&-50&-1&0\\& & \color{orangered}{-10} & & & \\ \hline &1&\color{orangered}{-5}&&& \end{array} $$

Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -10 } \cdot \color{blue}{ \left( -5 \right) } = \color{blue}{ 50 } $.

$$ \begin{array}{c|rrrrr}\color{blue}{-10}&1&5&-50&-1&0\\& & -10& \color{blue}{50} & & \\ \hline &1&\color{blue}{-5}&&& \end{array} $$

Step 5 : Add down last column: $ \color{orangered}{ -50 } + \color{orangered}{ 50 } = \color{orangered}{ 0 } $

$$ \begin{array}{c|rrrrr}-10&1&5&\color{orangered}{ -50 }&-1&0\\& & -10& \color{orangered}{50} & & \\ \hline &1&-5&\color{orangered}{0}&& \end{array} $$

Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -10 } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.

$$ \begin{array}{c|rrrrr}\color{blue}{-10}&1&5&-50&-1&0\\& & -10& 50& \color{blue}{0} & \\ \hline &1&-5&\color{blue}{0}&& \end{array} $$

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

$$ \begin{array}{c|rrrrr}-10&1&5&-50&\color{orangered}{ -1 }&0\\& & -10& 50& \color{orangered}{0} & \\ \hline &1&-5&0&\color{orangered}{-1}& \end{array} $$

Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -10 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ 10 } $.

$$ \begin{array}{c|rrrrr}\color{blue}{-10}&1&5&-50&-1&0\\& & -10& 50& 0& \color{blue}{10} \\ \hline &1&-5&0&\color{blue}{-1}& \end{array} $$

Step 9 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 10 } = \color{orangered}{ 10 } $

$$ \begin{array}{c|rrrrr}-10&1&5&-50&-1&\color{orangered}{ 0 }\\& & -10& 50& 0& \color{orangered}{10} \\ \hline &\color{blue}{1}&\color{blue}{-5}&\color{blue}{0}&\color{blue}{-1}&\color{orangered}{10} \end{array} $$

Remainder is the last entry in the bottom row $ \left(\color{red}{ 10 }\right) $.

Synthetic Division Calculator