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