Divide using synthetic division $$ ( 2x^{3}-13x^{2}+20x-4 ) \div ( x-2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}2&2&-13&20&-4\\& & 4& -18& \color{black}{4} \\ \hline &\color{blue}{2}&\color{blue}{-9}&\color{blue}{2}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ 2x^{3}-13x^{2}+20x-4 }{ x-2 } = \color{blue}{2x^{2}-9x+2} $$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.
$$ \begin{array}{c|rrrr}\color{blue}{2}&2&-13&20&-4\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}2&\color{orangered}{ 2 }&-13&20&-4\\& & & & \\ \hline &\color{orangered}{2}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&2&-13&20&-4\\& & \color{blue}{4} & & \\ \hline &\color{blue}{2}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -13 } + \color{orangered}{ 4 } = \color{orangered}{ -9 } $
$$ \begin{array}{c|rrrr}2&2&\color{orangered}{ -13 }&20&-4\\& & \color{orangered}{4} & & \\ \hline &2&\color{orangered}{-9}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -9 \right) } = \color{blue}{ -18 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&2&-13&20&-4\\& & 4& \color{blue}{-18} & \\ \hline &2&\color{blue}{-9}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 20 } + \color{orangered}{ \left( -18 \right) } = \color{orangered}{ 2 } $
$$ \begin{array}{c|rrrr}2&2&-13&\color{orangered}{ 20 }&-4\\& & 4& \color{orangered}{-18} & \\ \hline &2&-9&\color{orangered}{2}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 2 } = \color{blue}{ 4 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&2&-13&20&-4\\& & 4& -18& \color{blue}{4} \\ \hline &2&-9&\color{blue}{2}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -4 } + \color{orangered}{ 4 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}2&2&-13&20&\color{orangered}{ -4 }\\& & 4& -18& \color{orangered}{4} \\ \hline &\color{blue}{2}&\color{blue}{-9}&\color{blue}{2}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x^{2}-9x+2 } $ with a remainder of $ \color{red}{ 0 } $.