Divide using synthetic division $$ ( -9x^{3}+8x^{2}-7x+2 ) \div ( x-2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}2&-9&8&-7&2\\& & -18& -20& \color{black}{-54} \\ \hline &\color{blue}{-9}&\color{blue}{-10}&\color{blue}{-27}&\color{orangered}{-52} \end{array} $$The solution is:
$$ \frac{ -9x^{3}+8x^{2}-7x+2 }{ x-2 } = \color{blue}{-9x^{2}-10x-27} \color{red}{~-~} \frac{ \color{red}{ 52 } }{ x-2 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{2}&-9&8&-7&2\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}2&\color{orangered}{ -9 }&8&-7&2\\& & & & \\ \hline &\color{orangered}{-9}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -9 \right) } = \color{blue}{ -18 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&-9&8&-7&2\\& & \color{blue}{-18} & & \\ \hline &\color{blue}{-9}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 8 } + \color{orangered}{ \left( -18 \right) } = \color{orangered}{ -10 } $
$$ \begin{array}{c|rrrr}2&-9&\color{orangered}{ 8 }&-7&2\\& & \color{orangered}{-18} & & \\ \hline &-9&\color{orangered}{-10}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -10 \right) } = \color{blue}{ -20 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&-9&8&-7&2\\& & -18& \color{blue}{-20} & \\ \hline &-9&\color{blue}{-10}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -7 } + \color{orangered}{ \left( -20 \right) } = \color{orangered}{ -27 } $
$$ \begin{array}{c|rrrr}2&-9&8&\color{orangered}{ -7 }&2\\& & -18& \color{orangered}{-20} & \\ \hline &-9&-10&\color{orangered}{-27}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -27 \right) } = \color{blue}{ -54 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&-9&8&-7&2\\& & -18& -20& \color{blue}{-54} \\ \hline &-9&-10&\color{blue}{-27}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ \left( -54 \right) } = \color{orangered}{ -52 } $
$$ \begin{array}{c|rrrr}2&-9&8&-7&\color{orangered}{ 2 }\\& & -18& -20& \color{orangered}{-54} \\ \hline &\color{blue}{-9}&\color{blue}{-10}&\color{blue}{-27}&\color{orangered}{-52} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -9x^{2}-10x-27 } $ with a remainder of $ \color{red}{ -52 } $.