Divide using synthetic division $$ ( 5x^{6}+3x^{2}-200 ) \div ( x+5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrrr}-5&5&0&0&0&3&0&-200\\& & -25& 125& -625& 3125& -15640& \color{black}{78200} \\ \hline &\color{blue}{5}&\color{blue}{-25}&\color{blue}{125}&\color{blue}{-625}&\color{blue}{3128}&\color{blue}{-15640}&\color{orangered}{78000} \end{array} $$The solution is:
$$ \frac{ 5x^{6}+3x^{2}-200 }{ x+5 } = \color{blue}{5x^{5}-25x^{4}+125x^{3}-625x^{2}+3128x-15640} ~+~ \frac{ \color{red}{ 78000 } }{ x+5 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 5 = 0 $ ( $ x = \color{blue}{ -5 } $ ) at the left.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-5}&5&0&0&0&3&0&-200\\& & & & & & & \\ \hline &&&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrrr}-5&\color{orangered}{ 5 }&0&0&0&3&0&-200\\& & & & & & & \\ \hline &\color{orangered}{5}&&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 5 } = \color{blue}{ -25 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-5}&5&0&0&0&3&0&-200\\& & \color{blue}{-25} & & & & & \\ \hline &\color{blue}{5}&&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -25 \right) } = \color{orangered}{ -25 } $
$$ \begin{array}{c|rrrrrrr}-5&5&\color{orangered}{ 0 }&0&0&3&0&-200\\& & \color{orangered}{-25} & & & & & \\ \hline &5&\color{orangered}{-25}&&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -25 \right) } = \color{blue}{ 125 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-5}&5&0&0&0&3&0&-200\\& & -25& \color{blue}{125} & & & & \\ \hline &5&\color{blue}{-25}&&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 125 } = \color{orangered}{ 125 } $
$$ \begin{array}{c|rrrrrrr}-5&5&0&\color{orangered}{ 0 }&0&3&0&-200\\& & -25& \color{orangered}{125} & & & & \\ \hline &5&-25&\color{orangered}{125}&&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 125 } = \color{blue}{ -625 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-5}&5&0&0&0&3&0&-200\\& & -25& 125& \color{blue}{-625} & & & \\ \hline &5&-25&\color{blue}{125}&&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -625 \right) } = \color{orangered}{ -625 } $
$$ \begin{array}{c|rrrrrrr}-5&5&0&0&\color{orangered}{ 0 }&3&0&-200\\& & -25& 125& \color{orangered}{-625} & & & \\ \hline &5&-25&125&\color{orangered}{-625}&&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -625 \right) } = \color{blue}{ 3125 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-5}&5&0&0&0&3&0&-200\\& & -25& 125& -625& \color{blue}{3125} & & \\ \hline &5&-25&125&\color{blue}{-625}&&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ 3125 } = \color{orangered}{ 3128 } $
$$ \begin{array}{c|rrrrrrr}-5&5&0&0&0&\color{orangered}{ 3 }&0&-200\\& & -25& 125& -625& \color{orangered}{3125} & & \\ \hline &5&-25&125&-625&\color{orangered}{3128}&& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 3128 } = \color{blue}{ -15640 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-5}&5&0&0&0&3&0&-200\\& & -25& 125& -625& 3125& \color{blue}{-15640} & \\ \hline &5&-25&125&-625&\color{blue}{3128}&& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -15640 \right) } = \color{orangered}{ -15640 } $
$$ \begin{array}{c|rrrrrrr}-5&5&0&0&0&3&\color{orangered}{ 0 }&-200\\& & -25& 125& -625& 3125& \color{orangered}{-15640} & \\ \hline &5&-25&125&-625&3128&\color{orangered}{-15640}& \end{array} $$Step 12 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -15640 \right) } = \color{blue}{ 78200 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-5}&5&0&0&0&3&0&-200\\& & -25& 125& -625& 3125& -15640& \color{blue}{78200} \\ \hline &5&-25&125&-625&3128&\color{blue}{-15640}& \end{array} $$Step 13 : Add down last column: $ \color{orangered}{ -200 } + \color{orangered}{ 78200 } = \color{orangered}{ 78000 } $
$$ \begin{array}{c|rrrrrrr}-5&5&0&0&0&3&0&\color{orangered}{ -200 }\\& & -25& 125& -625& 3125& -15640& \color{orangered}{78200} \\ \hline &\color{blue}{5}&\color{blue}{-25}&\color{blue}{125}&\color{blue}{-625}&\color{blue}{3128}&\color{blue}{-15640}&\color{orangered}{78000} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 5x^{5}-25x^{4}+125x^{3}-625x^{2}+3128x-15640 } $ with a remainder of $ \color{red}{ 78000 } $.