Find remainder when $ x^{4}+3x^{3}+5x^{2}+10x-7 $ is divided by $ x+3 $.

The synthetic division table is:

$$ \begin{array}{c|rrrrr}-3&1&3&5&10&-7\\& & -3& 0& -15& \color{black}{15} \\ \hline &\color{blue}{1}&\color{blue}{0}&\color{blue}{5}&\color{blue}{-5}&\color{orangered}{8} \end{array} $$

The remainder when $ x^{4}+3x^{3}+5x^{2}+10x-7 $ is divided by $ x+3 $ is $ \, \color{red}{ 8 } $.

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.

$$ \begin{array}{c|rrrrr}\color{blue}{-3}&1&3&5&10&-7\\& & & & & \\ \hline &&&&& \end{array} $$

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

$$ \begin{array}{c|rrrrr}-3&\color{orangered}{ 1 }&3&5&10&-7\\& & & & & \\ \hline &\color{orangered}{1}&&&& \end{array} $$

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 } $.

$$ \begin{array}{c|rrrrr}\color{blue}{-3}&1&3&5&10&-7\\& & \color{blue}{-3} & & & \\ \hline &\color{blue}{1}&&&& \end{array} $$

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

$$ \begin{array}{c|rrrrr}-3&1&\color{orangered}{ 3 }&5&10&-7\\& & \color{orangered}{-3} & & & \\ \hline &1&\color{orangered}{0}&&& \end{array} $$

Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.

$$ \begin{array}{c|rrrrr}\color{blue}{-3}&1&3&5&10&-7\\& & -3& \color{blue}{0} & & \\ \hline &1&\color{blue}{0}&&& \end{array} $$

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

$$ \begin{array}{c|rrrrr}-3&1&3&\color{orangered}{ 5 }&10&-7\\& & -3& \color{orangered}{0} & & \\ \hline &1&0&\color{orangered}{5}&& \end{array} $$

Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 5 } = \color{blue}{ -15 } $.

$$ \begin{array}{c|rrrrr}\color{blue}{-3}&1&3&5&10&-7\\& & -3& 0& \color{blue}{-15} & \\ \hline &1&0&\color{blue}{5}&& \end{array} $$

Step 7 : Add down last column: $ \color{orangered}{ 10 } + \color{orangered}{ \left( -15 \right) } = \color{orangered}{ -5 } $

$$ \begin{array}{c|rrrrr}-3&1&3&5&\color{orangered}{ 10 }&-7\\& & -3& 0& \color{orangered}{-15} & \\ \hline &1&0&5&\color{orangered}{-5}& \end{array} $$

Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -5 \right) } = \color{blue}{ 15 } $.

$$ \begin{array}{c|rrrrr}\color{blue}{-3}&1&3&5&10&-7\\& & -3& 0& -15& \color{blue}{15} \\ \hline &1&0&5&\color{blue}{-5}& \end{array} $$

Step 9 : Add down last column: $ \color{orangered}{ -7 } + \color{orangered}{ 15 } = \color{orangered}{ 8 } $

$$ \begin{array}{c|rrrrr}-3&1&3&5&10&\color{orangered}{ -7 }\\& & -3& 0& -15& \color{orangered}{15} \\ \hline &\color{blue}{1}&\color{blue}{0}&\color{blue}{5}&\color{blue}{-5}&\color{orangered}{8} \end{array} $$

Remainder is the last entry in the bottom row $ \left(\color{red}{ 8 }\right) $.

Synthetic Division Calculator