Find remainder when $ x^{3}-34x-12 $ is divided by $ x+3 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-3&1&0&-34&-12\\& & -3& 9& \color{black}{75} \\ \hline &\color{blue}{1}&\color{blue}{-3}&\color{blue}{-25}&\color{orangered}{63} \end{array} $$The remainder when $ x^{3}-34x-12 $ is divided by $ x+3 $ is $ \, \color{red}{ 63 } $.
Explanation
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.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&1&0&-34&-12\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-3&\color{orangered}{ 1 }&0&-34&-12\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 1 } = \color{blue}{ -3 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&1&0&-34&-12\\& & \color{blue}{-3} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -3 \right) } = \color{orangered}{ -3 } $
$$ \begin{array}{c|rrrr}-3&1&\color{orangered}{ 0 }&-34&-12\\& & \color{orangered}{-3} & & \\ \hline &1&\color{orangered}{-3}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -3 \right) } = \color{blue}{ 9 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&1&0&-34&-12\\& & -3& \color{blue}{9} & \\ \hline &1&\color{blue}{-3}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -34 } + \color{orangered}{ 9 } = \color{orangered}{ -25 } $
$$ \begin{array}{c|rrrr}-3&1&0&\color{orangered}{ -34 }&-12\\& & -3& \color{orangered}{9} & \\ \hline &1&-3&\color{orangered}{-25}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -25 \right) } = \color{blue}{ 75 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&1&0&-34&-12\\& & -3& 9& \color{blue}{75} \\ \hline &1&-3&\color{blue}{-25}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -12 } + \color{orangered}{ 75 } = \color{orangered}{ 63 } $
$$ \begin{array}{c|rrrr}-3&1&0&-34&\color{orangered}{ -12 }\\& & -3& 9& \color{orangered}{75} \\ \hline &\color{blue}{1}&\color{blue}{-3}&\color{blue}{-25}&\color{orangered}{63} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 63 }\right) $.