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