Find remainder when $ 6x^{5}-29x^{4}-20x^{2}-28x+8 $ is divided by $ x-5 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}5&6&-29&0&-20&-28&8\\& & 30& 5& 25& 25& \color{black}{-15} \\ \hline &\color{blue}{6}&\color{blue}{1}&\color{blue}{5}&\color{blue}{5}&\color{blue}{-3}&\color{orangered}{-7} \end{array} $$The remainder when $ 6x^{5}-29x^{4}-20x^{2}-28x+8 $ is divided by $ x-5 $ is $ \, \color{red}{ -7 } $.
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 -5 = 0 $ ( $ x = \color{blue}{ 5 } $ ) at the left.
$$ \begin{array}{c|rrrrrr}\color{blue}{5}&6&-29&0&-20&-28&8\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}5&\color{orangered}{ 6 }&-29&0&-20&-28&8\\& & & & & & \\ \hline &\color{orangered}{6}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 6 } = \color{blue}{ 30 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{5}&6&-29&0&-20&-28&8\\& & \color{blue}{30} & & & & \\ \hline &\color{blue}{6}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -29 } + \color{orangered}{ 30 } = \color{orangered}{ 1 } $
$$ \begin{array}{c|rrrrrr}5&6&\color{orangered}{ -29 }&0&-20&-28&8\\& & \color{orangered}{30} & & & & \\ \hline &6&\color{orangered}{1}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 1 } = \color{blue}{ 5 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{5}&6&-29&0&-20&-28&8\\& & 30& \color{blue}{5} & & & \\ \hline &6&\color{blue}{1}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 5 } = \color{orangered}{ 5 } $
$$ \begin{array}{c|rrrrrr}5&6&-29&\color{orangered}{ 0 }&-20&-28&8\\& & 30& \color{orangered}{5} & & & \\ \hline &6&1&\color{orangered}{5}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 5 } = \color{blue}{ 25 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{5}&6&-29&0&-20&-28&8\\& & 30& 5& \color{blue}{25} & & \\ \hline &6&1&\color{blue}{5}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -20 } + \color{orangered}{ 25 } = \color{orangered}{ 5 } $
$$ \begin{array}{c|rrrrrr}5&6&-29&0&\color{orangered}{ -20 }&-28&8\\& & 30& 5& \color{orangered}{25} & & \\ \hline &6&1&5&\color{orangered}{5}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 5 } = \color{blue}{ 25 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{5}&6&-29&0&-20&-28&8\\& & 30& 5& 25& \color{blue}{25} & \\ \hline &6&1&5&\color{blue}{5}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -28 } + \color{orangered}{ 25 } = \color{orangered}{ -3 } $
$$ \begin{array}{c|rrrrrr}5&6&-29&0&-20&\color{orangered}{ -28 }&8\\& & 30& 5& 25& \color{orangered}{25} & \\ \hline &6&1&5&5&\color{orangered}{-3}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ \left( -3 \right) } = \color{blue}{ -15 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{5}&6&-29&0&-20&-28&8\\& & 30& 5& 25& 25& \color{blue}{-15} \\ \hline &6&1&5&5&\color{blue}{-3}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 8 } + \color{orangered}{ \left( -15 \right) } = \color{orangered}{ -7 } $
$$ \begin{array}{c|rrrrrr}5&6&-29&0&-20&-28&\color{orangered}{ 8 }\\& & 30& 5& 25& 25& \color{orangered}{-15} \\ \hline &\color{blue}{6}&\color{blue}{1}&\color{blue}{5}&\color{blue}{5}&\color{blue}{-3}&\color{orangered}{-7} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ -7 }\right) $.