Find remainder when $ -20x^{2}+11x-26 $ is divided by $ x-2 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}2&-20&11&-26\\& & -40& \color{black}{-58} \\ \hline &\color{blue}{-20}&\color{blue}{-29}&\color{orangered}{-84} \end{array} $$The remainder when $ -20x^{2}+11x-26 $ is divided by $ x-2 $ is $ \, \color{red}{ -84 } $.
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.
$$ \begin{array}{c|rrr}\color{blue}{2}&-20&11&-26\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}2&\color{orangered}{ -20 }&11&-26\\& & & \\ \hline &\color{orangered}{-20}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -20 \right) } = \color{blue}{ -40 } $.
$$ \begin{array}{c|rrr}\color{blue}{2}&-20&11&-26\\& & \color{blue}{-40} & \\ \hline &\color{blue}{-20}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 11 } + \color{orangered}{ \left( -40 \right) } = \color{orangered}{ -29 } $
$$ \begin{array}{c|rrr}2&-20&\color{orangered}{ 11 }&-26\\& & \color{orangered}{-40} & \\ \hline &-20&\color{orangered}{-29}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -29 \right) } = \color{blue}{ -58 } $.
$$ \begin{array}{c|rrr}\color{blue}{2}&-20&11&-26\\& & -40& \color{blue}{-58} \\ \hline &-20&\color{blue}{-29}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -26 } + \color{orangered}{ \left( -58 \right) } = \color{orangered}{ -84 } $
$$ \begin{array}{c|rrr}2&-20&11&\color{orangered}{ -26 }\\& & -40& \color{orangered}{-58} \\ \hline &\color{blue}{-20}&\color{blue}{-29}&\color{orangered}{-84} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ -84 }\right) $.