Find remainder when $ 10x^{5}-9x^{4}-17x^{3}+36 $ is divided by $ x-2 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}2&10&-9&-17&0&0&36\\& & 20& 22& 10& 20& \color{black}{40} \\ \hline &\color{blue}{10}&\color{blue}{11}&\color{blue}{5}&\color{blue}{10}&\color{blue}{20}&\color{orangered}{76} \end{array} $$The remainder when $ 10x^{5}-9x^{4}-17x^{3}+36 $ is divided by $ x-2 $ is $ \, \color{red}{ 76 } $.
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 -2 = 0 $ ( $ x = \color{blue}{ 2 } $ ) at the left.
$$ \begin{array}{c|rrrrrr}\color{blue}{2}&10&-9&-17&0&0&36\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}2&\color{orangered}{ 10 }&-9&-17&0&0&36\\& & & & & & \\ \hline &\color{orangered}{10}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 10 } = \color{blue}{ 20 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{2}&10&-9&-17&0&0&36\\& & \color{blue}{20} & & & & \\ \hline &\color{blue}{10}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -9 } + \color{orangered}{ 20 } = \color{orangered}{ 11 } $
$$ \begin{array}{c|rrrrrr}2&10&\color{orangered}{ -9 }&-17&0&0&36\\& & \color{orangered}{20} & & & & \\ \hline &10&\color{orangered}{11}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 11 } = \color{blue}{ 22 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{2}&10&-9&-17&0&0&36\\& & 20& \color{blue}{22} & & & \\ \hline &10&\color{blue}{11}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -17 } + \color{orangered}{ 22 } = \color{orangered}{ 5 } $
$$ \begin{array}{c|rrrrrr}2&10&-9&\color{orangered}{ -17 }&0&0&36\\& & 20& \color{orangered}{22} & & & \\ \hline &10&11&\color{orangered}{5}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 5 } = \color{blue}{ 10 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{2}&10&-9&-17&0&0&36\\& & 20& 22& \color{blue}{10} & & \\ \hline &10&11&\color{blue}{5}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 10 } = \color{orangered}{ 10 } $
$$ \begin{array}{c|rrrrrr}2&10&-9&-17&\color{orangered}{ 0 }&0&36\\& & 20& 22& \color{orangered}{10} & & \\ \hline &10&11&5&\color{orangered}{10}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 10 } = \color{blue}{ 20 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{2}&10&-9&-17&0&0&36\\& & 20& 22& 10& \color{blue}{20} & \\ \hline &10&11&5&\color{blue}{10}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 20 } = \color{orangered}{ 20 } $
$$ \begin{array}{c|rrrrrr}2&10&-9&-17&0&\color{orangered}{ 0 }&36\\& & 20& 22& 10& \color{orangered}{20} & \\ \hline &10&11&5&10&\color{orangered}{20}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 20 } = \color{blue}{ 40 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{2}&10&-9&-17&0&0&36\\& & 20& 22& 10& 20& \color{blue}{40} \\ \hline &10&11&5&10&\color{blue}{20}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 36 } + \color{orangered}{ 40 } = \color{orangered}{ 76 } $
$$ \begin{array}{c|rrrrrr}2&10&-9&-17&0&0&\color{orangered}{ 36 }\\& & 20& 22& 10& 20& \color{orangered}{40} \\ \hline &\color{blue}{10}&\color{blue}{11}&\color{blue}{5}&\color{blue}{10}&\color{blue}{20}&\color{orangered}{76} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 76 }\right) $.