Find remainder when $ 2x^{4}-4x^{3}+20x-100 $ is divided by $ x+11 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-11&2&-4&0&20&-100\\& & -22& 286& -3146& \color{black}{34386} \\ \hline &\color{blue}{2}&\color{blue}{-26}&\color{blue}{286}&\color{blue}{-3126}&\color{orangered}{34286} \end{array} $$The remainder when $ 2x^{4}-4x^{3}+20x-100 $ is divided by $ x+11 $ is $ \, \color{red}{ 34286 } $.
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 + 11 = 0 $ ( $ x = \color{blue}{ -11 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{-11}&2&-4&0&20&-100\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-11&\color{orangered}{ 2 }&-4&0&20&-100\\& & & & & \\ \hline &\color{orangered}{2}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -11 } \cdot \color{blue}{ 2 } = \color{blue}{ -22 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-11}&2&-4&0&20&-100\\& & \color{blue}{-22} & & & \\ \hline &\color{blue}{2}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -4 } + \color{orangered}{ \left( -22 \right) } = \color{orangered}{ -26 } $
$$ \begin{array}{c|rrrrr}-11&2&\color{orangered}{ -4 }&0&20&-100\\& & \color{orangered}{-22} & & & \\ \hline &2&\color{orangered}{-26}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -11 } \cdot \color{blue}{ \left( -26 \right) } = \color{blue}{ 286 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-11}&2&-4&0&20&-100\\& & -22& \color{blue}{286} & & \\ \hline &2&\color{blue}{-26}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 286 } = \color{orangered}{ 286 } $
$$ \begin{array}{c|rrrrr}-11&2&-4&\color{orangered}{ 0 }&20&-100\\& & -22& \color{orangered}{286} & & \\ \hline &2&-26&\color{orangered}{286}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -11 } \cdot \color{blue}{ 286 } = \color{blue}{ -3146 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-11}&2&-4&0&20&-100\\& & -22& 286& \color{blue}{-3146} & \\ \hline &2&-26&\color{blue}{286}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 20 } + \color{orangered}{ \left( -3146 \right) } = \color{orangered}{ -3126 } $
$$ \begin{array}{c|rrrrr}-11&2&-4&0&\color{orangered}{ 20 }&-100\\& & -22& 286& \color{orangered}{-3146} & \\ \hline &2&-26&286&\color{orangered}{-3126}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -11 } \cdot \color{blue}{ \left( -3126 \right) } = \color{blue}{ 34386 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-11}&2&-4&0&20&-100\\& & -22& 286& -3146& \color{blue}{34386} \\ \hline &2&-26&286&\color{blue}{-3126}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -100 } + \color{orangered}{ 34386 } = \color{orangered}{ 34286 } $
$$ \begin{array}{c|rrrrr}-11&2&-4&0&20&\color{orangered}{ -100 }\\& & -22& 286& -3146& \color{orangered}{34386} \\ \hline &\color{blue}{2}&\color{blue}{-26}&\color{blue}{286}&\color{blue}{-3126}&\color{orangered}{34286} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 34286 }\right) $.