Divide using synthetic division $$ ( -x^{3}+5x^{2}-11x+10 ) \div ( x-2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}2&-1&5&-11&10\\& & -2& 6& \color{black}{-10} \\ \hline &\color{blue}{-1}&\color{blue}{3}&\color{blue}{-5}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ -x^{3}+5x^{2}-11x+10 }{ x-2 } = \color{blue}{-x^{2}+3x-5} $$Explanation
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.
$$ \begin{array}{c|rrrr}\color{blue}{2}&-1&5&-11&10\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}2&\color{orangered}{ -1 }&5&-11&10\\& & & & \\ \hline &\color{orangered}{-1}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ -2 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&-1&5&-11&10\\& & \color{blue}{-2} & & \\ \hline &\color{blue}{-1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ \left( -2 \right) } = \color{orangered}{ 3 } $
$$ \begin{array}{c|rrrr}2&-1&\color{orangered}{ 5 }&-11&10\\& & \color{orangered}{-2} & & \\ \hline &-1&\color{orangered}{3}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 3 } = \color{blue}{ 6 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&-1&5&-11&10\\& & -2& \color{blue}{6} & \\ \hline &-1&\color{blue}{3}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -11 } + \color{orangered}{ 6 } = \color{orangered}{ -5 } $
$$ \begin{array}{c|rrrr}2&-1&5&\color{orangered}{ -11 }&10\\& & -2& \color{orangered}{6} & \\ \hline &-1&3&\color{orangered}{-5}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -5 \right) } = \color{blue}{ -10 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&-1&5&-11&10\\& & -2& 6& \color{blue}{-10} \\ \hline &-1&3&\color{blue}{-5}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 10 } + \color{orangered}{ \left( -10 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}2&-1&5&-11&\color{orangered}{ 10 }\\& & -2& 6& \color{orangered}{-10} \\ \hline &\color{blue}{-1}&\color{blue}{3}&\color{blue}{-5}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -x^{2}+3x-5 } $ with a remainder of $ \color{red}{ 0 } $.