Find remainder when $ 13x^{3}-x^{2}-221x+17 $ is divided by $ x+1 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-1&13&-1&-221&17\\& & -13& 14& \color{black}{207} \\ \hline &\color{blue}{13}&\color{blue}{-14}&\color{blue}{-207}&\color{orangered}{224} \end{array} $$The remainder when $ 13x^{3}-x^{2}-221x+17 $ is divided by $ x+1 $ is $ \, \color{red}{ 224 } $.
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}{ \left( -13 \right) } = \color{orangered}{ -14 } $
$$ \begin{array}{c|rrrr}-1&13&\color{orangered}{ -1 }&-221&17\\& & \color{orangered}{-13} & & \\ \hline &13&\color{orangered}{-14}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -14 \right) } = \color{blue}{ 14 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&13&-1&-221&17\\& & -13& \color{blue}{14} & \\ \hline &13&\color{blue}{-14}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -221 } + \color{orangered}{ 14 } = \color{orangered}{ -207 } $
$$ \begin{array}{c|rrrr}-1&13&-1&\color{orangered}{ -221 }&17\\& & -13& \color{orangered}{14} & \\ \hline &13&-14&\color{orangered}{-207}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -207 \right) } = \color{blue}{ 207 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&13&-1&-221&17\\& & -13& 14& \color{blue}{207} \\ \hline &13&-14&\color{blue}{-207}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 17 } + \color{orangered}{ 207 } = \color{orangered}{ 224 } $
$$ \begin{array}{c|rrrr}-1&13&-1&-221&\color{orangered}{ 17 }\\& & -13& 14& \color{orangered}{207} \\ \hline &\color{blue}{13}&\color{blue}{-14}&\color{blue}{-207}&\color{orangered}{224} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 224 }\right) $.