Find remainder when $ x^{2}+5x+1 $ is divided by $ x+3 $.

The synthetic division table is:

$$ \begin{array}{c|rrr}-3&1&5&1\\& & -3& \color{black}{-6} \\ \hline &\color{blue}{1}&\color{blue}{2}&\color{orangered}{-5} \end{array} $$

The remainder when $ x^{2}+5x+1 $ is divided by $ x+3 $ is $ \, \color{red}{ -5 } $.

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 + 3 = 0 $ ( $ x = \color{blue}{ -3 } $ ) at the left.

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

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

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

Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 1 } = \color{blue}{ -3 } $.

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

Step 3 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ \left( -3 \right) } = \color{orangered}{ 2 } $

$$ \begin{array}{c|rrr}-3&1&\color{orangered}{ 5 }&1\\& & \color{orangered}{-3} & \\ \hline &1&\color{orangered}{2}& \end{array} $$

Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 2 } = \color{blue}{ -6 } $.

$$ \begin{array}{c|rrr}\color{blue}{-3}&1&5&1\\& & -3& \color{blue}{-6} \\ \hline &1&\color{blue}{2}& \end{array} $$

Step 5 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ \left( -6 \right) } = \color{orangered}{ -5 } $

$$ \begin{array}{c|rrr}-3&1&5&\color{orangered}{ 1 }\\& & -3& \color{orangered}{-6} \\ \hline &\color{blue}{1}&\color{blue}{2}&\color{orangered}{-5} \end{array} $$

Remainder is the last entry in the bottom row $ \left(\color{red}{ -5 }\right) $.

Synthetic Division Calculator