Divide using synthetic division $$ ( 1043.3561x^{4}-15647.056x^{2}+58089.6544 ) \div ( 64.62x ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}0&\frac{ 10433561 }{ 10000 }&0&-\frac{ 1955882 }{ 125 }&0&\frac{ 36306034 }{ 625 }\\& & 0& 0& 0& \color{black}{0} \\ \hline &\color{blue}{\frac{ 10433561 }{ 10000 }}&\color{blue}{0}&\color{blue}{-\frac{ 1955882 }{ 125 }}&\color{blue}{0}&\color{orangered}{\frac{ 36306034 }{ 625 }} \end{array} $$The solution is:
$$ \frac{ \frac{ 10433561 }{ 10000 }x^{4}-\frac{ 1955882 }{ 125 }x^{2}+\frac{ 36306034 }{ 625 } }{ x } = \color{blue}{\frac{ 10433561 }{ 10000 }x^{3}-\frac{ 1955882 }{ 125 }x} ~+~ \frac{ \color{red}{ \frac{ 36306034 }{ 625 } } }{ x } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table.Put the zero at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{0}&\frac{ 10433561 }{ 10000 }&0&-\frac{ 1955882 }{ 125 }&0&\frac{ 36306034 }{ 625 }\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}0&\color{orangered}{ \frac{ 10433561 }{ 10000 } }&0&-\frac{ 1955882 }{ 125 }&0&\frac{ 36306034 }{ 625 }\\& & & & & \\ \hline &\color{orangered}{\frac{ 10433561 }{ 10000 }}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ \frac{ 10433561 }{ 10000 } } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{0}&\frac{ 10433561 }{ 10000 }&0&-\frac{ 1955882 }{ 125 }&0&\frac{ 36306034 }{ 625 }\\& & \color{blue}{0} & & & \\ \hline &\color{blue}{\frac{ 10433561 }{ 10000 }}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 0 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrr}0&\frac{ 10433561 }{ 10000 }&\color{orangered}{ 0 }&-\frac{ 1955882 }{ 125 }&0&\frac{ 36306034 }{ 625 }\\& & \color{orangered}{0} & & & \\ \hline &\frac{ 10433561 }{ 10000 }&\color{orangered}{0}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{0}&\frac{ 10433561 }{ 10000 }&0&-\frac{ 1955882 }{ 125 }&0&\frac{ 36306034 }{ 625 }\\& & 0& \color{blue}{0} & & \\ \hline &\frac{ 10433561 }{ 10000 }&\color{blue}{0}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -\frac{ 1955882 }{ 125 } } + \color{orangered}{ 0 } = \color{orangered}{ -\frac{ 1955882 }{ 125 } } $
$$ \begin{array}{c|rrrrr}0&\frac{ 10433561 }{ 10000 }&0&\color{orangered}{ -\frac{ 1955882 }{ 125 } }&0&\frac{ 36306034 }{ 625 }\\& & 0& \color{orangered}{0} & & \\ \hline &\frac{ 10433561 }{ 10000 }&0&\color{orangered}{-\frac{ 1955882 }{ 125 }}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ \left( -\frac{ 1955882 }{ 125 } \right) } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{0}&\frac{ 10433561 }{ 10000 }&0&-\frac{ 1955882 }{ 125 }&0&\frac{ 36306034 }{ 625 }\\& & 0& 0& \color{blue}{0} & \\ \hline &\frac{ 10433561 }{ 10000 }&0&\color{blue}{-\frac{ 1955882 }{ 125 }}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 0 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrr}0&\frac{ 10433561 }{ 10000 }&0&-\frac{ 1955882 }{ 125 }&\color{orangered}{ 0 }&\frac{ 36306034 }{ 625 }\\& & 0& 0& \color{orangered}{0} & \\ \hline &\frac{ 10433561 }{ 10000 }&0&-\frac{ 1955882 }{ 125 }&\color{orangered}{0}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{0}&\frac{ 10433561 }{ 10000 }&0&-\frac{ 1955882 }{ 125 }&0&\frac{ 36306034 }{ 625 }\\& & 0& 0& 0& \color{blue}{0} \\ \hline &\frac{ 10433561 }{ 10000 }&0&-\frac{ 1955882 }{ 125 }&\color{blue}{0}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ \frac{ 36306034 }{ 625 } } + \color{orangered}{ 0 } = \color{orangered}{ \frac{ 36306034 }{ 625 } } $
$$ \begin{array}{c|rrrrr}0&\frac{ 10433561 }{ 10000 }&0&-\frac{ 1955882 }{ 125 }&0&\color{orangered}{ \frac{ 36306034 }{ 625 } }\\& & 0& 0& 0& \color{orangered}{0} \\ \hline &\color{blue}{\frac{ 10433561 }{ 10000 }}&\color{blue}{0}&\color{blue}{-\frac{ 1955882 }{ 125 }}&\color{blue}{0}&\color{orangered}{\frac{ 36306034 }{ 625 }} \end{array} $$Bottom line represents the quotient $ \color{blue}{ \frac{ 10433561 }{ 10000 }x^{3}-\frac{ 1955882 }{ 125 }x } $ with a remainder of $ \color{red}{ \frac{ 36306034 }{ 625 } } $.