Divide using synthetic division $$ ( 2x^{5}-2x^{3}+4x^{2}-3 ) \div ( 1 ) $$

The synthetic division table is:

$$ \begin{array}{c|rrrrrr}0&2&0&-2&4&0&-3\\& & 0& 0& 0& 0& \color{black}{0} \\ \hline &\color{blue}{2}&\color{blue}{0}&\color{blue}{-2}&\color{blue}{4}&\color{blue}{0}&\color{orangered}{-3} \end{array} $$

The solution is:

$$ \frac{ 2x^{5}-2x^{3}+4x^{2}-3 }{ x } = \color{blue}{2x^{4}-2x^{2}+4x} \color{red}{~-~} \frac{ \color{red}{ 3 } }{ x } $$

Step 1 : Write down the coefficients of the dividend into division table.Put the zero at the left.

$$ \begin{array}{c|rrrrrr}\color{blue}{0}&2&0&-2&4&0&-3\\& & & & & & \\ \hline &&&&&& \end{array} $$

Step 1 : Bring down the leading coefficient to the bottom row.

$$ \begin{array}{c|rrrrrr}0&\color{orangered}{ 2 }&0&-2&4&0&-3\\& & & & & & \\ \hline &\color{orangered}{2}&&&&& \end{array} $$

Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 2 } = \color{blue}{ 0 } $.

$$ \begin{array}{c|rrrrrr}\color{blue}{0}&2&0&-2&4&0&-3\\& & \color{blue}{0} & & & & \\ \hline &\color{blue}{2}&&&&& \end{array} $$

Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 0 } = \color{orangered}{ 0 } $

$$ \begin{array}{c|rrrrrr}0&2&\color{orangered}{ 0 }&-2&4&0&-3\\& & \color{orangered}{0} & & & & \\ \hline &2&\color{orangered}{0}&&&& \end{array} $$

Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.

$$ \begin{array}{c|rrrrrr}\color{blue}{0}&2&0&-2&4&0&-3\\& & 0& \color{blue}{0} & & & \\ \hline &2&\color{blue}{0}&&&& \end{array} $$

Step 5 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ 0 } = \color{orangered}{ -2 } $

$$ \begin{array}{c|rrrrrr}0&2&0&\color{orangered}{ -2 }&4&0&-3\\& & 0& \color{orangered}{0} & & & \\ \hline &2&0&\color{orangered}{-2}&&& \end{array} $$

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 } $.

$$ \begin{array}{c|rrrrrr}\color{blue}{0}&2&0&-2&4&0&-3\\& & 0& 0& \color{blue}{0} & & \\ \hline &2&0&\color{blue}{-2}&&& \end{array} $$

Step 7 : Add down last column: $ \color{orangered}{ 4 } + \color{orangered}{ 0 } = \color{orangered}{ 4 } $

$$ \begin{array}{c|rrrrrr}0&2&0&-2&\color{orangered}{ 4 }&0&-3\\& & 0& 0& \color{orangered}{0} & & \\ \hline &2&0&-2&\color{orangered}{4}&& \end{array} $$

Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 4 } = \color{blue}{ 0 } $.

$$ \begin{array}{c|rrrrrr}\color{blue}{0}&2&0&-2&4&0&-3\\& & 0& 0& 0& \color{blue}{0} & \\ \hline &2&0&-2&\color{blue}{4}&& \end{array} $$

Step 9 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 0 } = \color{orangered}{ 0 } $

$$ \begin{array}{c|rrrrrr}0&2&0&-2&4&\color{orangered}{ 0 }&-3\\& & 0& 0& 0& \color{orangered}{0} & \\ \hline &2&0&-2&4&\color{orangered}{0}& \end{array} $$

Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.

$$ \begin{array}{c|rrrrrr}\color{blue}{0}&2&0&-2&4&0&-3\\& & 0& 0& 0& 0& \color{blue}{0} \\ \hline &2&0&-2&4&\color{blue}{0}& \end{array} $$

Step 11 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ 0 } = \color{orangered}{ -3 } $

$$ \begin{array}{c|rrrrrr}0&2&0&-2&4&0&\color{orangered}{ -3 }\\& & 0& 0& 0& 0& \color{orangered}{0} \\ \hline &\color{blue}{2}&\color{blue}{0}&\color{blue}{-2}&\color{blue}{4}&\color{blue}{0}&\color{orangered}{-3} \end{array} $$

Bottom line represents the quotient $ \color{blue}{ 2x^{4}-2x^{2}+4x } $ with a remainder of $ \color{red}{ -3 } $.

Synthetic Division Calculator