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