Find remainder when $ 2x^{3}-54 $ is divided by $ x-3 $.

The synthetic division table is:

$$ \begin{array}{c|rrrr}3&2&0&0&-54\\& & 6& 18& \color{black}{54} \\ \hline &\color{blue}{2}&\color{blue}{6}&\color{blue}{18}&\color{orangered}{0} \end{array} $$

The remainder when $ 2x^{3}-54 $ is divided by $ x-3 $ is $ \, \color{red}{ 0 } $.

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.

$$ \begin{array}{c|rrrr}\color{blue}{3}&2&0&0&-54\\& & & & \\ \hline &&&& \end{array} $$

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

$$ \begin{array}{c|rrrr}3&\color{orangered}{ 2 }&0&0&-54\\& & & & \\ \hline &\color{orangered}{2}&&& \end{array} $$

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

$$ \begin{array}{c|rrrr}\color{blue}{3}&2&0&0&-54\\& & \color{blue}{6} & & \\ \hline &\color{blue}{2}&&& \end{array} $$

Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 6 } = \color{orangered}{ 6 } $

$$ \begin{array}{c|rrrr}3&2&\color{orangered}{ 0 }&0&-54\\& & \color{orangered}{6} & & \\ \hline &2&\color{orangered}{6}&& \end{array} $$

Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 6 } = \color{blue}{ 18 } $.

$$ \begin{array}{c|rrrr}\color{blue}{3}&2&0&0&-54\\& & 6& \color{blue}{18} & \\ \hline &2&\color{blue}{6}&& \end{array} $$

Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 18 } = \color{orangered}{ 18 } $

$$ \begin{array}{c|rrrr}3&2&0&\color{orangered}{ 0 }&-54\\& & 6& \color{orangered}{18} & \\ \hline &2&6&\color{orangered}{18}& \end{array} $$

Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 18 } = \color{blue}{ 54 } $.

$$ \begin{array}{c|rrrr}\color{blue}{3}&2&0&0&-54\\& & 6& 18& \color{blue}{54} \\ \hline &2&6&\color{blue}{18}& \end{array} $$

Step 7 : Add down last column: $ \color{orangered}{ -54 } + \color{orangered}{ 54 } = \color{orangered}{ 0 } $

$$ \begin{array}{c|rrrr}3&2&0&0&\color{orangered}{ -54 }\\& & 6& 18& \color{orangered}{54} \\ \hline &\color{blue}{2}&\color{blue}{6}&\color{blue}{18}&\color{orangered}{0} \end{array} $$

Remainder is the last entry in the bottom row $ \left(\color{red}{ 0 }\right) $.

Synthetic Division Calculator