Find remainder when $ 2x^{5}-3x^{4}-9x^{3}+8x^{2}+2 $ is divided by $ -3 $.

The synthetic division table is:

$$ \begin{array}{c|rrrrrr}0&2&-3&-9&8&0&2\\& & 0& 0& 0& 0& \color{black}{0} \\ \hline &\color{blue}{2}&\color{blue}{-3}&\color{blue}{-9}&\color{blue}{8}&\color{blue}{0}&\color{orangered}{2} \end{array} $$

The remainder when $ 2x^{5}-3x^{4}-9x^{3}+8x^{2}+2 $ is divided by $ x $ is $ \, \color{red}{ 2 } $.

We can find remainder using synthetic division method.

Step 1 : Write down the coefficients of the dividend into division table.Put the zero at the left.

$$ \begin{array}{c|rrrrrr}\color{blue}{0}&2&-3&-9&8&0&2\\& & & & & & \\ \hline &&&&&& \end{array} $$

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

$$ \begin{array}{c|rrrrrr}0&\color{orangered}{ 2 }&-3&-9&8&0&2\\& & & & & & \\ \hline &\color{orangered}{2}&&&&& \end{array} $$

Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 2 } = \color{blue}{ 0 } $.

$$ \begin{array}{c|rrrrrr}\color{blue}{0}&2&-3&-9&8&0&2\\& & \color{blue}{0} & & & & \\ \hline &\color{blue}{2}&&&&& \end{array} $$

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

$$ \begin{array}{c|rrrrrr}0&2&\color{orangered}{ -3 }&-9&8&0&2\\& & \color{orangered}{0} & & & & \\ \hline &2&\color{orangered}{-3}&&&& \end{array} $$

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

$$ \begin{array}{c|rrrrrr}\color{blue}{0}&2&-3&-9&8&0&2\\& & 0& \color{blue}{0} & & & \\ \hline &2&\color{blue}{-3}&&&& \end{array} $$

Step 5 : Add down last column: $ \color{orangered}{ -9 } + \color{orangered}{ 0 } = \color{orangered}{ -9 } $

$$ \begin{array}{c|rrrrrr}0&2&-3&\color{orangered}{ -9 }&8&0&2\\& & 0& \color{orangered}{0} & & & \\ \hline &2&-3&\color{orangered}{-9}&&& \end{array} $$

Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ \left( -9 \right) } = \color{blue}{ 0 } $.

$$ \begin{array}{c|rrrrrr}\color{blue}{0}&2&-3&-9&8&0&2\\& & 0& 0& \color{blue}{0} & & \\ \hline &2&-3&\color{blue}{-9}&&& \end{array} $$

Step 7 : Add down last column: $ \color{orangered}{ 8 } + \color{orangered}{ 0 } = \color{orangered}{ 8 } $

$$ \begin{array}{c|rrrrrr}0&2&-3&-9&\color{orangered}{ 8 }&0&2\\& & 0& 0& \color{orangered}{0} & & \\ \hline &2&-3&-9&\color{orangered}{8}&& \end{array} $$

Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 8 } = \color{blue}{ 0 } $.

$$ \begin{array}{c|rrrrrr}\color{blue}{0}&2&-3&-9&8&0&2\\& & 0& 0& 0& \color{blue}{0} & \\ \hline &2&-3&-9&\color{blue}{8}&& \end{array} $$

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

$$ \begin{array}{c|rrrrrr}0&2&-3&-9&8&\color{orangered}{ 0 }&2\\& & 0& 0& 0& \color{orangered}{0} & \\ \hline &2&-3&-9&8&\color{orangered}{0}& \end{array} $$

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

$$ \begin{array}{c|rrrrrr}\color{blue}{0}&2&-3&-9&8&0&2\\& & 0& 0& 0& 0& \color{blue}{0} \\ \hline &2&-3&-9&8&\color{blue}{0}& \end{array} $$

Step 11 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ 0 } = \color{orangered}{ 2 } $

$$ \begin{array}{c|rrrrrr}0&2&-3&-9&8&0&\color{orangered}{ 2 }\\& & 0& 0& 0& 0& \color{orangered}{0} \\ \hline &\color{blue}{2}&\color{blue}{-3}&\color{blue}{-9}&\color{blue}{8}&\color{blue}{0}&\color{orangered}{2} \end{array} $$

Remainder is the last entry in the bottom row $ \left(\color{red}{ 2 }\right) $.

Synthetic Division Calculator