Find remainder when $ 13x^{3}-x^{2}-221x+17 $ is divided by $ x-1 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}1&13&-1&-221&17\\& & 13& 12& \color{black}{-209} \\ \hline &\color{blue}{13}&\color{blue}{12}&\color{blue}{-209}&\color{orangered}{-192} \end{array} $$The remainder when $ 13x^{3}-x^{2}-221x+17 $ is divided by $ x-1 $ is $ \, \color{red}{ -192 } $.
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 -1 = 0 $ ( $ x = \color{blue}{ 1 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{1}&13&-1&-221&17\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}1&\color{orangered}{ 13 }&-1&-221&17\\& & & & \\ \hline &\color{orangered}{13}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 13 } = \color{blue}{ 13 } $.
$$ \begin{array}{c|rrrr}\color{blue}{1}&13&-1&-221&17\\& & \color{blue}{13} & & \\ \hline &\color{blue}{13}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ 13 } = \color{orangered}{ 12 } $
$$ \begin{array}{c|rrrr}1&13&\color{orangered}{ -1 }&-221&17\\& & \color{orangered}{13} & & \\ \hline &13&\color{orangered}{12}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 12 } = \color{blue}{ 12 } $.
$$ \begin{array}{c|rrrr}\color{blue}{1}&13&-1&-221&17\\& & 13& \color{blue}{12} & \\ \hline &13&\color{blue}{12}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -221 } + \color{orangered}{ 12 } = \color{orangered}{ -209 } $
$$ \begin{array}{c|rrrr}1&13&-1&\color{orangered}{ -221 }&17\\& & 13& \color{orangered}{12} & \\ \hline &13&12&\color{orangered}{-209}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ \left( -209 \right) } = \color{blue}{ -209 } $.
$$ \begin{array}{c|rrrr}\color{blue}{1}&13&-1&-221&17\\& & 13& 12& \color{blue}{-209} \\ \hline &13&12&\color{blue}{-209}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 17 } + \color{orangered}{ \left( -209 \right) } = \color{orangered}{ -192 } $
$$ \begin{array}{c|rrrr}1&13&-1&-221&\color{orangered}{ 17 }\\& & 13& 12& \color{orangered}{-209} \\ \hline &\color{blue}{13}&\color{blue}{12}&\color{blue}{-209}&\color{orangered}{-192} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ -192 }\right) $.