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