Divide using synthetic division $$ ( 10x^{3}-20x^{2}+4x ) \div ( x-2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}2&10&-20&4&0\\& & 20& 0& \color{black}{8} \\ \hline &\color{blue}{10}&\color{blue}{0}&\color{blue}{4}&\color{orangered}{8} \end{array} $$The solution is:
$$ \frac{ 10x^{3}-20x^{2}+4x }{ x-2 } = \color{blue}{10x^{2}+4} ~+~ \frac{ \color{red}{ 8 } }{ x-2 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{2}&10&-20&4&0\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}2&\color{orangered}{ 10 }&-20&4&0\\& & & & \\ \hline &\color{orangered}{10}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 10 } = \color{blue}{ 20 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&10&-20&4&0\\& & \color{blue}{20} & & \\ \hline &\color{blue}{10}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -20 } + \color{orangered}{ 20 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}2&10&\color{orangered}{ -20 }&4&0\\& & \color{orangered}{20} & & \\ \hline &10&\color{orangered}{0}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&10&-20&4&0\\& & 20& \color{blue}{0} & \\ \hline &10&\color{blue}{0}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 4 } + \color{orangered}{ 0 } = \color{orangered}{ 4 } $
$$ \begin{array}{c|rrrr}2&10&-20&\color{orangered}{ 4 }&0\\& & 20& \color{orangered}{0} & \\ \hline &10&0&\color{orangered}{4}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 4 } = \color{blue}{ 8 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&10&-20&4&0\\& & 20& 0& \color{blue}{8} \\ \hline &10&0&\color{blue}{4}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 8 } = \color{orangered}{ 8 } $
$$ \begin{array}{c|rrrr}2&10&-20&4&\color{orangered}{ 0 }\\& & 20& 0& \color{orangered}{8} \\ \hline &\color{blue}{10}&\color{blue}{0}&\color{blue}{4}&\color{orangered}{8} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 10x^{2}+4 } $ with a remainder of $ \color{red}{ 8 } $.