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