Divide using synthetic division $$ ( 13x^{4}+23x^{3}+18x^{2}+21x ) \div ( 2 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrrrr}0&13&23&18&21&0\\& & 0& 0& 0& \color{black}{0} \\ \hline &\color{blue}{13}&\color{blue}{23}&\color{blue}{18}&\color{blue}{21}&\color{orangered}{0} \end{array} $$The solution is:
$$ \dfrac{ 13x^{4}+23x^{3}+18x^{2}+21x }{ x } = \color{blue}{13x^{3}+23x^{2}+18x+21} $$Step 1 : Write down the coefficients of the dividend into division table.Put the zero at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{0}&13&23&18&21&0\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}0&\color{orangered}{ 13 }&23&18&21&0\\& & & & & \\ \hline &\color{orangered}{13}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 13 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{0}&13&23&18&21&0\\& & \color{blue}{0} & & & \\ \hline &\color{blue}{13}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 23 } + \color{orangered}{ 0 } = \color{orangered}{ 23 } $
$$ \begin{array}{c|rrrrr}0&13&\color{orangered}{ 23 }&18&21&0\\& & \color{orangered}{0} & & & \\ \hline &13&\color{orangered}{23}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 23 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{0}&13&23&18&21&0\\& & 0& \color{blue}{0} & & \\ \hline &13&\color{blue}{23}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 18 } + \color{orangered}{ 0 } = \color{orangered}{ 18 } $
$$ \begin{array}{c|rrrrr}0&13&23&\color{orangered}{ 18 }&21&0\\& & 0& \color{orangered}{0} & & \\ \hline &13&23&\color{orangered}{18}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 18 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{0}&13&23&18&21&0\\& & 0& 0& \color{blue}{0} & \\ \hline &13&23&\color{blue}{18}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 21 } + \color{orangered}{ 0 } = \color{orangered}{ 21 } $
$$ \begin{array}{c|rrrrr}0&13&23&18&\color{orangered}{ 21 }&0\\& & 0& 0& \color{orangered}{0} & \\ \hline &13&23&18&\color{orangered}{21}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 21 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{0}&13&23&18&21&0\\& & 0& 0& 0& \color{blue}{0} \\ \hline &13&23&18&\color{blue}{21}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 0 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrr}0&13&23&18&21&\color{orangered}{ 0 }\\& & 0& 0& 0& \color{orangered}{0} \\ \hline &\color{blue}{13}&\color{blue}{23}&\color{blue}{18}&\color{blue}{21}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 13x^{3}+23x^{2}+18x+21 } $ with a remainder of $ \color{red}{ 0 } $.