Divide using synthetic division $$ ( -x^{3}+12x^{2}-19x+8 ) \div ( x-1 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}1&-1&12&-19&8\\& & -1& 11& \color{black}{-8} \\ \hline &\color{blue}{-1}&\color{blue}{11}&\color{blue}{-8}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ -x^{3}+12x^{2}-19x+8 }{ x-1 } = \color{blue}{-x^{2}+11x-8} $$Explanation
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.
$$ \begin{array}{c|rrrr}\color{blue}{1}&-1&12&-19&8\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}1&\color{orangered}{ -1 }&12&-19&8\\& & & & \\ \hline &\color{orangered}{-1}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ -1 } $.
$$ \begin{array}{c|rrrr}\color{blue}{1}&-1&12&-19&8\\& & \color{blue}{-1} & & \\ \hline &\color{blue}{-1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 12 } + \color{orangered}{ \left( -1 \right) } = \color{orangered}{ 11 } $
$$ \begin{array}{c|rrrr}1&-1&\color{orangered}{ 12 }&-19&8\\& & \color{orangered}{-1} & & \\ \hline &-1&\color{orangered}{11}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 11 } = \color{blue}{ 11 } $.
$$ \begin{array}{c|rrrr}\color{blue}{1}&-1&12&-19&8\\& & -1& \color{blue}{11} & \\ \hline &-1&\color{blue}{11}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -19 } + \color{orangered}{ 11 } = \color{orangered}{ -8 } $
$$ \begin{array}{c|rrrr}1&-1&12&\color{orangered}{ -19 }&8\\& & -1& \color{orangered}{11} & \\ \hline &-1&11&\color{orangered}{-8}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ \left( -8 \right) } = \color{blue}{ -8 } $.
$$ \begin{array}{c|rrrr}\color{blue}{1}&-1&12&-19&8\\& & -1& 11& \color{blue}{-8} \\ \hline &-1&11&\color{blue}{-8}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 8 } + \color{orangered}{ \left( -8 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}1&-1&12&-19&\color{orangered}{ 8 }\\& & -1& 11& \color{orangered}{-8} \\ \hline &\color{blue}{-1}&\color{blue}{11}&\color{blue}{-8}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -x^{2}+11x-8 } $ with a remainder of $ \color{red}{ 0 } $.