Divide using synthetic division $$ ( 4x^{5}+x^{4}+4x^{2}+5 ) \div ( -2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}0&4&1&0&4&0&5\\& & 0& 0& 0& 0& \color{black}{0} \\ \hline &\color{blue}{4}&\color{blue}{1}&\color{blue}{0}&\color{blue}{4}&\color{blue}{0}&\color{orangered}{5} \end{array} $$The solution is:
$$ \frac{ 4x^{5}+x^{4}+4x^{2}+5 }{ x } = \color{blue}{4x^{4}+x^{3}+4x} ~+~ \frac{ \color{red}{ 5 } }{ x } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table.Put the zero at the left.
$$ \begin{array}{c|rrrrrr}\color{blue}{0}&4&1&0&4&0&5\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}0&\color{orangered}{ 4 }&1&0&4&0&5\\& & & & & & \\ \hline &\color{orangered}{4}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 4 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{0}&4&1&0&4&0&5\\& & \color{blue}{0} & & & & \\ \hline &\color{blue}{4}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ 0 } = \color{orangered}{ 1 } $
$$ \begin{array}{c|rrrrrr}0&4&\color{orangered}{ 1 }&0&4&0&5\\& & \color{orangered}{0} & & & & \\ \hline &4&\color{orangered}{1}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 1 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{0}&4&1&0&4&0&5\\& & 0& \color{blue}{0} & & & \\ \hline &4&\color{blue}{1}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 0 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrrr}0&4&1&\color{orangered}{ 0 }&4&0&5\\& & 0& \color{orangered}{0} & & & \\ \hline &4&1&\color{orangered}{0}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{0}&4&1&0&4&0&5\\& & 0& 0& \color{blue}{0} & & \\ \hline &4&1&\color{blue}{0}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 4 } + \color{orangered}{ 0 } = \color{orangered}{ 4 } $
$$ \begin{array}{c|rrrrrr}0&4&1&0&\color{orangered}{ 4 }&0&5\\& & 0& 0& \color{orangered}{0} & & \\ \hline &4&1&0&\color{orangered}{4}&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{0}&4&1&0&4&0&5\\& & 0& 0& 0& \color{blue}{0} & \\ \hline &4&1&0&\color{blue}{4}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 0 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrrr}0&4&1&0&4&\color{orangered}{ 0 }&5\\& & 0& 0& 0& \color{orangered}{0} & \\ \hline &4&1&0&4&\color{orangered}{0}& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{0}&4&1&0&4&0&5\\& & 0& 0& 0& 0& \color{blue}{0} \\ \hline &4&1&0&4&\color{blue}{0}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ 0 } = \color{orangered}{ 5 } $
$$ \begin{array}{c|rrrrrr}0&4&1&0&4&0&\color{orangered}{ 5 }\\& & 0& 0& 0& 0& \color{orangered}{0} \\ \hline &\color{blue}{4}&\color{blue}{1}&\color{blue}{0}&\color{blue}{4}&\color{blue}{0}&\color{orangered}{5} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 4x^{4}+x^{3}+4x } $ with a remainder of $ \color{red}{ 5 } $.