Find remainder when $ x^{3}-12x^{2}+44x-48 $ is divided by $ x-6 $.
The synthetic division table is:
$$ \begin{array}{c|rrrr}6&1&-12&44&-48\\& & 6& -36& \color{black}{48} \\ \hline &\color{blue}{1}&\color{blue}{-6}&\color{blue}{8}&\color{orangered}{0} \end{array} $$The remainder when $ x^{3}-12x^{2}+44x-48 $ is divided by $ x-6 $ 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 -6 = 0 $ ( $ x = \color{blue}{ 6 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{6}&1&-12&44&-48\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}6&\color{orangered}{ 1 }&-12&44&-48\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ 1 } = \color{blue}{ 6 } $.
$$ \begin{array}{c|rrrr}\color{blue}{6}&1&-12&44&-48\\& & \color{blue}{6} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -12 } + \color{orangered}{ 6 } = \color{orangered}{ -6 } $
$$ \begin{array}{c|rrrr}6&1&\color{orangered}{ -12 }&44&-48\\& & \color{orangered}{6} & & \\ \hline &1&\color{orangered}{-6}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ \left( -6 \right) } = \color{blue}{ -36 } $.
$$ \begin{array}{c|rrrr}\color{blue}{6}&1&-12&44&-48\\& & 6& \color{blue}{-36} & \\ \hline &1&\color{blue}{-6}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 44 } + \color{orangered}{ \left( -36 \right) } = \color{orangered}{ 8 } $
$$ \begin{array}{c|rrrr}6&1&-12&\color{orangered}{ 44 }&-48\\& & 6& \color{orangered}{-36} & \\ \hline &1&-6&\color{orangered}{8}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ 8 } = \color{blue}{ 48 } $.
$$ \begin{array}{c|rrrr}\color{blue}{6}&1&-12&44&-48\\& & 6& -36& \color{blue}{48} \\ \hline &1&-6&\color{blue}{8}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -48 } + \color{orangered}{ 48 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}6&1&-12&44&\color{orangered}{ -48 }\\& & 6& -36& \color{orangered}{48} \\ \hline &\color{blue}{1}&\color{blue}{-6}&\color{blue}{8}&\color{orangered}{0} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 0 }\right) $.