Divide using synthetic division $$ ( 3x^{5}+13x^{4}+35x^{3}+105x^{2}+72x-108 ) \div ( -2x ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}0&3&13&35&105&72&-108\\& & 0& 0& 0& 0& \color{black}{0} \\ \hline &\color{blue}{3}&\color{blue}{13}&\color{blue}{35}&\color{blue}{105}&\color{blue}{72}&\color{orangered}{-108} \end{array} $$The solution is:
$$ \frac{ 3x^{5}+13x^{4}+35x^{3}+105x^{2}+72x-108 }{ x } = \color{blue}{3x^{4}+13x^{3}+35x^{2}+105x+72} \color{red}{~-~} \frac{ \color{red}{ 108 } }{ x } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table.Put the zero at the left.
$$ \begin{array}{c|rrrrrr}\color{blue}{0}&3&13&35&105&72&-108\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}0&\color{orangered}{ 3 }&13&35&105&72&-108\\& & & & & & \\ \hline &\color{orangered}{3}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 3 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{0}&3&13&35&105&72&-108\\& & \color{blue}{0} & & & & \\ \hline &\color{blue}{3}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 13 } + \color{orangered}{ 0 } = \color{orangered}{ 13 } $
$$ \begin{array}{c|rrrrrr}0&3&\color{orangered}{ 13 }&35&105&72&-108\\& & \color{orangered}{0} & & & & \\ \hline &3&\color{orangered}{13}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 13 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{0}&3&13&35&105&72&-108\\& & 0& \color{blue}{0} & & & \\ \hline &3&\color{blue}{13}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 35 } + \color{orangered}{ 0 } = \color{orangered}{ 35 } $
$$ \begin{array}{c|rrrrrr}0&3&13&\color{orangered}{ 35 }&105&72&-108\\& & 0& \color{orangered}{0} & & & \\ \hline &3&13&\color{orangered}{35}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 35 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{0}&3&13&35&105&72&-108\\& & 0& 0& \color{blue}{0} & & \\ \hline &3&13&\color{blue}{35}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 105 } + \color{orangered}{ 0 } = \color{orangered}{ 105 } $
$$ \begin{array}{c|rrrrrr}0&3&13&35&\color{orangered}{ 105 }&72&-108\\& & 0& 0& \color{orangered}{0} & & \\ \hline &3&13&35&\color{orangered}{105}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 105 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{0}&3&13&35&105&72&-108\\& & 0& 0& 0& \color{blue}{0} & \\ \hline &3&13&35&\color{blue}{105}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 72 } + \color{orangered}{ 0 } = \color{orangered}{ 72 } $
$$ \begin{array}{c|rrrrrr}0&3&13&35&105&\color{orangered}{ 72 }&-108\\& & 0& 0& 0& \color{orangered}{0} & \\ \hline &3&13&35&105&\color{orangered}{72}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 72 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{0}&3&13&35&105&72&-108\\& & 0& 0& 0& 0& \color{blue}{0} \\ \hline &3&13&35&105&\color{blue}{72}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ -108 } + \color{orangered}{ 0 } = \color{orangered}{ -108 } $
$$ \begin{array}{c|rrrrrr}0&3&13&35&105&72&\color{orangered}{ -108 }\\& & 0& 0& 0& 0& \color{orangered}{0} \\ \hline &\color{blue}{3}&\color{blue}{13}&\color{blue}{35}&\color{blue}{105}&\color{blue}{72}&\color{orangered}{-108} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{4}+13x^{3}+35x^{2}+105x+72 } $ with a remainder of $ \color{red}{ -108 } $.