Find remainder when $ 14x^{4}-25x^{3}+3x^{2}-5x+36 $ is divided by $ x-2 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}2&14&-25&3&-5&36\\& & 28& 6& 18& \color{black}{26} \\ \hline &\color{blue}{14}&\color{blue}{3}&\color{blue}{9}&\color{blue}{13}&\color{orangered}{62} \end{array} $$The remainder when $ 14x^{4}-25x^{3}+3x^{2}-5x+36 $ is divided by $ x-2 $ is $ \, \color{red}{ 62 } $.
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.
$$ \begin{array}{c|rrrrr}\color{blue}{2}&14&-25&3&-5&36\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}2&\color{orangered}{ 14 }&-25&3&-5&36\\& & & & & \\ \hline &\color{orangered}{14}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 14 } = \color{blue}{ 28 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{2}&14&-25&3&-5&36\\& & \color{blue}{28} & & & \\ \hline &\color{blue}{14}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -25 } + \color{orangered}{ 28 } = \color{orangered}{ 3 } $
$$ \begin{array}{c|rrrrr}2&14&\color{orangered}{ -25 }&3&-5&36\\& & \color{orangered}{28} & & & \\ \hline &14&\color{orangered}{3}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 3 } = \color{blue}{ 6 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{2}&14&-25&3&-5&36\\& & 28& \color{blue}{6} & & \\ \hline &14&\color{blue}{3}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ 6 } = \color{orangered}{ 9 } $
$$ \begin{array}{c|rrrrr}2&14&-25&\color{orangered}{ 3 }&-5&36\\& & 28& \color{orangered}{6} & & \\ \hline &14&3&\color{orangered}{9}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 9 } = \color{blue}{ 18 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{2}&14&-25&3&-5&36\\& & 28& 6& \color{blue}{18} & \\ \hline &14&3&\color{blue}{9}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -5 } + \color{orangered}{ 18 } = \color{orangered}{ 13 } $
$$ \begin{array}{c|rrrrr}2&14&-25&3&\color{orangered}{ -5 }&36\\& & 28& 6& \color{orangered}{18} & \\ \hline &14&3&9&\color{orangered}{13}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 13 } = \color{blue}{ 26 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{2}&14&-25&3&-5&36\\& & 28& 6& 18& \color{blue}{26} \\ \hline &14&3&9&\color{blue}{13}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 36 } + \color{orangered}{ 26 } = \color{orangered}{ 62 } $
$$ \begin{array}{c|rrrrr}2&14&-25&3&-5&\color{orangered}{ 36 }\\& & 28& 6& 18& \color{orangered}{26} \\ \hline &\color{blue}{14}&\color{blue}{3}&\color{blue}{9}&\color{blue}{13}&\color{orangered}{62} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 62 }\right) $.