Find remainder when $ x^{3}+5x^{2}-4x+6 $ is divided by $ x+5 $.

The synthetic division table is:

$$ \begin{array}{c|rrrr}-5&1&5&-4&6\\& & -5& 0& \color{black}{20} \\ \hline &\color{blue}{1}&\color{blue}{0}&\color{blue}{-4}&\color{orangered}{26} \end{array} $$

The remainder when $ x^{3}+5x^{2}-4x+6 $ is divided by $ x+5 $ is $ \, \color{red}{ 26 } $.

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.

$$ \begin{array}{c|rrrr}\color{blue}{-5}&1&5&-4&6\\& & & & \\ \hline &&&& \end{array} $$

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

$$ \begin{array}{c|rrrr}-5&\color{orangered}{ 1 }&5&-4&6\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$

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 } $.

$$ \begin{array}{c|rrrr}\color{blue}{-5}&1&5&-4&6\\& & \color{blue}{-5} & & \\ \hline &\color{blue}{1}&&& \end{array} $$

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

$$ \begin{array}{c|rrrr}-5&1&\color{orangered}{ 5 }&-4&6\\& & \color{orangered}{-5} & & \\ \hline &1&\color{orangered}{0}&& \end{array} $$

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 } $.

$$ \begin{array}{c|rrrr}\color{blue}{-5}&1&5&-4&6\\& & -5& \color{blue}{0} & \\ \hline &1&\color{blue}{0}&& \end{array} $$

Step 5 : Add down last column: $ \color{orangered}{ -4 } + \color{orangered}{ 0 } = \color{orangered}{ -4 } $

$$ \begin{array}{c|rrrr}-5&1&5&\color{orangered}{ -4 }&6\\& & -5& \color{orangered}{0} & \\ \hline &1&0&\color{orangered}{-4}& \end{array} $$

Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -4 \right) } = \color{blue}{ 20 } $.

$$ \begin{array}{c|rrrr}\color{blue}{-5}&1&5&-4&6\\& & -5& 0& \color{blue}{20} \\ \hline &1&0&\color{blue}{-4}& \end{array} $$

Step 7 : Add down last column: $ \color{orangered}{ 6 } + \color{orangered}{ 20 } = \color{orangered}{ 26 } $

$$ \begin{array}{c|rrrr}-5&1&5&-4&\color{orangered}{ 6 }\\& & -5& 0& \color{orangered}{20} \\ \hline &\color{blue}{1}&\color{blue}{0}&\color{blue}{-4}&\color{orangered}{26} \end{array} $$

Remainder is the last entry in the bottom row $ \left(\color{red}{ 26 }\right) $.

Synthetic Division Calculator