Find remainder when $ x^{3}-34x-12 $ is divided by $ x+12 $.
The synthetic division table is:
$$ \begin{array}{c|rrrr}-12&1&0&-34&-12\\& & -12& 144& \color{black}{-1320} \\ \hline &\color{blue}{1}&\color{blue}{-12}&\color{blue}{110}&\color{orangered}{-1332} \end{array} $$The remainder when $ x^{3}-34x-12 $ is divided by $ x+12 $ is $ \, \color{red}{ -1332 } $.
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 + 12 = 0 $ ( $ x = \color{blue}{ -12 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{-12}&1&0&-34&-12\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-12&\color{orangered}{ 1 }&0&-34&-12\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -12 } \cdot \color{blue}{ 1 } = \color{blue}{ -12 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-12}&1&0&-34&-12\\& & \color{blue}{-12} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -12 \right) } = \color{orangered}{ -12 } $
$$ \begin{array}{c|rrrr}-12&1&\color{orangered}{ 0 }&-34&-12\\& & \color{orangered}{-12} & & \\ \hline &1&\color{orangered}{-12}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -12 } \cdot \color{blue}{ \left( -12 \right) } = \color{blue}{ 144 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-12}&1&0&-34&-12\\& & -12& \color{blue}{144} & \\ \hline &1&\color{blue}{-12}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -34 } + \color{orangered}{ 144 } = \color{orangered}{ 110 } $
$$ \begin{array}{c|rrrr}-12&1&0&\color{orangered}{ -34 }&-12\\& & -12& \color{orangered}{144} & \\ \hline &1&-12&\color{orangered}{110}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -12 } \cdot \color{blue}{ 110 } = \color{blue}{ -1320 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-12}&1&0&-34&-12\\& & -12& 144& \color{blue}{-1320} \\ \hline &1&-12&\color{blue}{110}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -12 } + \color{orangered}{ \left( -1320 \right) } = \color{orangered}{ -1332 } $
$$ \begin{array}{c|rrrr}-12&1&0&-34&\color{orangered}{ -12 }\\& & -12& 144& \color{orangered}{-1320} \\ \hline &\color{blue}{1}&\color{blue}{-12}&\color{blue}{110}&\color{orangered}{-1332} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ -1332 }\right) $.