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

The synthetic division table is:

$$ \begin{array}{c|rrrrrr}-5&1&0&0&0&0&0\\& & -5& 25& -125& 625& \color{black}{-3125} \\ \hline &\color{blue}{1}&\color{blue}{-5}&\color{blue}{25}&\color{blue}{-125}&\color{blue}{625}&\color{orangered}{-3125} \end{array} $$

The remainder when $ x^{5} $ is divided by $ x+5 $ is $ \, \color{red}{ -3125 } $.

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.

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

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

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

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

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

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

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

Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -5 \right) } = \color{blue}{ 25 } $.

$$ \begin{array}{c|rrrrrr}\color{blue}{-5}&1&0&0&0&0&0\\& & -5& \color{blue}{25} & & & \\ \hline &1&\color{blue}{-5}&&&& \end{array} $$

Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 25 } = \color{orangered}{ 25 } $

$$ \begin{array}{c|rrrrrr}-5&1&0&\color{orangered}{ 0 }&0&0&0\\& & -5& \color{orangered}{25} & & & \\ \hline &1&-5&\color{orangered}{25}&&& \end{array} $$

Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 25 } = \color{blue}{ -125 } $.

$$ \begin{array}{c|rrrrrr}\color{blue}{-5}&1&0&0&0&0&0\\& & -5& 25& \color{blue}{-125} & & \\ \hline &1&-5&\color{blue}{25}&&& \end{array} $$

Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -125 \right) } = \color{orangered}{ -125 } $

$$ \begin{array}{c|rrrrrr}-5&1&0&0&\color{orangered}{ 0 }&0&0\\& & -5& 25& \color{orangered}{-125} & & \\ \hline &1&-5&25&\color{orangered}{-125}&& \end{array} $$

Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -125 \right) } = \color{blue}{ 625 } $.

$$ \begin{array}{c|rrrrrr}\color{blue}{-5}&1&0&0&0&0&0\\& & -5& 25& -125& \color{blue}{625} & \\ \hline &1&-5&25&\color{blue}{-125}&& \end{array} $$

Step 9 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 625 } = \color{orangered}{ 625 } $

$$ \begin{array}{c|rrrrrr}-5&1&0&0&0&\color{orangered}{ 0 }&0\\& & -5& 25& -125& \color{orangered}{625} & \\ \hline &1&-5&25&-125&\color{orangered}{625}& \end{array} $$

Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 625 } = \color{blue}{ -3125 } $.

$$ \begin{array}{c|rrrrrr}\color{blue}{-5}&1&0&0&0&0&0\\& & -5& 25& -125& 625& \color{blue}{-3125} \\ \hline &1&-5&25&-125&\color{blue}{625}& \end{array} $$

Step 11 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -3125 \right) } = \color{orangered}{ -3125 } $

$$ \begin{array}{c|rrrrrr}-5&1&0&0&0&0&\color{orangered}{ 0 }\\& & -5& 25& -125& 625& \color{orangered}{-3125} \\ \hline &\color{blue}{1}&\color{blue}{-5}&\color{blue}{25}&\color{blue}{-125}&\color{blue}{625}&\color{orangered}{-3125} \end{array} $$

Remainder is the last entry in the bottom row $ \left(\color{red}{ -3125 }\right) $.

Synthetic Division Calculator