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

The synthetic division table is:

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

The remainder when $ -4x^{3}+10x^{2}-6 $ is divided by $ x-2 $ 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 from $ x -2 = 0 $ ( $ x = \color{blue}{ 2 } $ ) at the left.

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

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

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

Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -4 \right) } = \color{blue}{ -8 } $.

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

Step 3 : Add down last column: $ \color{orangered}{ 10 } + \color{orangered}{ \left( -8 \right) } = \color{orangered}{ 2 } $

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

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

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

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

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

Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 4 } = \color{blue}{ 8 } $.

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

Step 7 : Add down last column: $ \color{orangered}{ -6 } + \color{orangered}{ 8 } = \color{orangered}{ 2 } $

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

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

Synthetic Division Calculator