Divide using synthetic division $$ ( 2x^{3}-5x^{2}-9x+18 ) \div ( x-2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}2&2&-5&-9&18\\& & 4& -2& \color{black}{-22} \\ \hline &\color{blue}{2}&\color{blue}{-1}&\color{blue}{-11}&\color{orangered}{-4} \end{array} $$The solution is:
$$ \frac{ 2x^{3}-5x^{2}-9x+18 }{ x-2 } = \color{blue}{2x^{2}-x-11} \color{red}{~-~} \frac{ \color{red}{ 4 } }{ x-2 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{2}&2&-5&-9&18\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}2&\color{orangered}{ 2 }&-5&-9&18\\& & & & \\ \hline &\color{orangered}{2}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 2 } = \color{blue}{ 4 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&2&-5&-9&18\\& & \color{blue}{4} & & \\ \hline &\color{blue}{2}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -5 } + \color{orangered}{ 4 } = \color{orangered}{ -1 } $
$$ \begin{array}{c|rrrr}2&2&\color{orangered}{ -5 }&-9&18\\& & \color{orangered}{4} & & \\ \hline &2&\color{orangered}{-1}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ -2 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&2&-5&-9&18\\& & 4& \color{blue}{-2} & \\ \hline &2&\color{blue}{-1}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -9 } + \color{orangered}{ \left( -2 \right) } = \color{orangered}{ -11 } $
$$ \begin{array}{c|rrrr}2&2&-5&\color{orangered}{ -9 }&18\\& & 4& \color{orangered}{-2} & \\ \hline &2&-1&\color{orangered}{-11}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -11 \right) } = \color{blue}{ -22 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&2&-5&-9&18\\& & 4& -2& \color{blue}{-22} \\ \hline &2&-1&\color{blue}{-11}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 18 } + \color{orangered}{ \left( -22 \right) } = \color{orangered}{ -4 } $
$$ \begin{array}{c|rrrr}2&2&-5&-9&\color{orangered}{ 18 }\\& & 4& -2& \color{orangered}{-22} \\ \hline &\color{blue}{2}&\color{blue}{-1}&\color{blue}{-11}&\color{orangered}{-4} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x^{2}-x-11 } $ with a remainder of $ \color{red}{ -4 } $.