Find remainder when $ x^{5}+6x^{4}-3x^{2}-22x-29 $ is divided by $ x+6 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}-6&1&6&0&-3&-22&-29\\& & -6& 0& 0& 18& \color{black}{24} \\ \hline &\color{blue}{1}&\color{blue}{0}&\color{blue}{0}&\color{blue}{-3}&\color{blue}{-4}&\color{orangered}{-5} \end{array} $$The remainder when $ x^{5}+6x^{4}-3x^{2}-22x-29 $ is divided by $ x+6 $ is $ \, \color{red}{ -5 } $.
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 + 6 = 0 $ ( $ x = \color{blue}{ -6 } $ ) at the left.
$$ \begin{array}{c|rrrrrr}\color{blue}{-6}&1&6&0&-3&-22&-29\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}-6&\color{orangered}{ 1 }&6&0&-3&-22&-29\\& & & & & & \\ \hline &\color{orangered}{1}&&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-6}&1&6&0&-3&-22&-29\\& & \color{blue}{-6} & & & & \\ \hline &\color{blue}{1}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 6 } + \color{orangered}{ \left( -6 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrrr}-6&1&\color{orangered}{ 6 }&0&-3&-22&-29\\& & \color{orangered}{-6} & & & & \\ \hline &1&\color{orangered}{0}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-6}&1&6&0&-3&-22&-29\\& & -6& \color{blue}{0} & & & \\ \hline &1&\color{blue}{0}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 0 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrrr}-6&1&6&\color{orangered}{ 0 }&-3&-22&-29\\& & -6& \color{orangered}{0} & & & \\ \hline &1&0&\color{orangered}{0}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-6}&1&6&0&-3&-22&-29\\& & -6& 0& \color{blue}{0} & & \\ \hline &1&0&\color{blue}{0}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ 0 } = \color{orangered}{ -3 } $
$$ \begin{array}{c|rrrrrr}-6&1&6&0&\color{orangered}{ -3 }&-22&-29\\& & -6& 0& \color{orangered}{0} & & \\ \hline &1&0&0&\color{orangered}{-3}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ \left( -3 \right) } = \color{blue}{ 18 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-6}&1&6&0&-3&-22&-29\\& & -6& 0& 0& \color{blue}{18} & \\ \hline &1&0&0&\color{blue}{-3}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -22 } + \color{orangered}{ 18 } = \color{orangered}{ -4 } $
$$ \begin{array}{c|rrrrrr}-6&1&6&0&-3&\color{orangered}{ -22 }&-29\\& & -6& 0& 0& \color{orangered}{18} & \\ \hline &1&0&0&-3&\color{orangered}{-4}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ \left( -4 \right) } = \color{blue}{ 24 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-6}&1&6&0&-3&-22&-29\\& & -6& 0& 0& 18& \color{blue}{24} \\ \hline &1&0&0&-3&\color{blue}{-4}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ -29 } + \color{orangered}{ 24 } = \color{orangered}{ -5 } $
$$ \begin{array}{c|rrrrrr}-6&1&6&0&-3&-22&\color{orangered}{ -29 }\\& & -6& 0& 0& 18& \color{orangered}{24} \\ \hline &\color{blue}{1}&\color{blue}{0}&\color{blue}{0}&\color{blue}{-3}&\color{blue}{-4}&\color{orangered}{-5} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ -5 }\right) $.