Find remainder when $ 4x^{5}-13x^{4}+8x^{3}-10x^{2}+6x+2 $ is divided by $ 0 $.

The synthetic division table is:

$$ \begin{array}{c|rrrrrr}0&4&-13&8&-10&6&2\\& & 0& 0& 0& 0& \color{black}{0} \\ \hline &\color{blue}{4}&\color{blue}{-13}&\color{blue}{8}&\color{blue}{-10}&\color{blue}{6}&\color{orangered}{2} \end{array} $$

The remainder when $ 4x^{5}-13x^{4}+8x^{3}-10x^{2}+6x+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.

$$ \begin{array}{c|rrrrrr}\color{blue}{0}&4&-13&8&-10&6&2\\& & & & & & \\ \hline &&&&&& \end{array} $$

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

$$ \begin{array}{c|rrrrrr}0&\color{orangered}{ 4 }&-13&8&-10&6&2\\& & & & & & \\ \hline &\color{orangered}{4}&&&&& \end{array} $$

Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 4 } = \color{blue}{ 0 } $.

$$ \begin{array}{c|rrrrrr}\color{blue}{0}&4&-13&8&-10&6&2\\& & \color{blue}{0} & & & & \\ \hline &\color{blue}{4}&&&&& \end{array} $$

Step 3 : Add down last column: $ \color{orangered}{ -13 } + \color{orangered}{ 0 } = \color{orangered}{ -13 } $

$$ \begin{array}{c|rrrrrr}0&4&\color{orangered}{ -13 }&8&-10&6&2\\& & \color{orangered}{0} & & & & \\ \hline &4&\color{orangered}{-13}&&&& \end{array} $$

Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ \left( -13 \right) } = \color{blue}{ 0 } $.

$$ \begin{array}{c|rrrrrr}\color{blue}{0}&4&-13&8&-10&6&2\\& & 0& \color{blue}{0} & & & \\ \hline &4&\color{blue}{-13}&&&& \end{array} $$

Step 5 : Add down last column: $ \color{orangered}{ 8 } + \color{orangered}{ 0 } = \color{orangered}{ 8 } $

$$ \begin{array}{c|rrrrrr}0&4&-13&\color{orangered}{ 8 }&-10&6&2\\& & 0& \color{orangered}{0} & & & \\ \hline &4&-13&\color{orangered}{8}&&& \end{array} $$

Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 8 } = \color{blue}{ 0 } $.

$$ \begin{array}{c|rrrrrr}\color{blue}{0}&4&-13&8&-10&6&2\\& & 0& 0& \color{blue}{0} & & \\ \hline &4&-13&\color{blue}{8}&&& \end{array} $$

Step 7 : Add down last column: $ \color{orangered}{ -10 } + \color{orangered}{ 0 } = \color{orangered}{ -10 } $

$$ \begin{array}{c|rrrrrr}0&4&-13&8&\color{orangered}{ -10 }&6&2\\& & 0& 0& \color{orangered}{0} & & \\ \hline &4&-13&8&\color{orangered}{-10}&& \end{array} $$

Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ \left( -10 \right) } = \color{blue}{ 0 } $.

$$ \begin{array}{c|rrrrrr}\color{blue}{0}&4&-13&8&-10&6&2\\& & 0& 0& 0& \color{blue}{0} & \\ \hline &4&-13&8&\color{blue}{-10}&& \end{array} $$

Step 9 : Add down last column: $ \color{orangered}{ 6 } + \color{orangered}{ 0 } = \color{orangered}{ 6 } $

$$ \begin{array}{c|rrrrrr}0&4&-13&8&-10&\color{orangered}{ 6 }&2\\& & 0& 0& 0& \color{orangered}{0} & \\ \hline &4&-13&8&-10&\color{orangered}{6}& \end{array} $$

Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 6 } = \color{blue}{ 0 } $.

$$ \begin{array}{c|rrrrrr}\color{blue}{0}&4&-13&8&-10&6&2\\& & 0& 0& 0& 0& \color{blue}{0} \\ \hline &4&-13&8&-10&\color{blue}{6}& \end{array} $$

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

$$ \begin{array}{c|rrrrrr}0&4&-13&8&-10&6&\color{orangered}{ 2 }\\& & 0& 0& 0& 0& \color{orangered}{0} \\ \hline &\color{blue}{4}&\color{blue}{-13}&\color{blue}{8}&\color{blue}{-10}&\color{blue}{6}&\color{orangered}{2} \end{array} $$

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

Synthetic Division Calculator