Divide using synthetic division $$ ( -10x^{4}+1250x+25 ) \div ( x-5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}5&-10&0&0&1250&25\\& & -50& -250& -1250& \color{black}{0} \\ \hline &\color{blue}{-10}&\color{blue}{-50}&\color{blue}{-250}&\color{blue}{0}&\color{orangered}{25} \end{array} $$The solution is:
$$ \frac{ -10x^{4}+1250x+25 }{ x-5 } = \color{blue}{-10x^{3}-50x^{2}-250x} ~+~ \frac{ \color{red}{ 25 } }{ x-5 } $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{5}&-10&0&0&1250&25\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}5&\color{orangered}{ -10 }&0&0&1250&25\\& & & & & \\ \hline &\color{orangered}{-10}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ \left( -10 \right) } = \color{blue}{ -50 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{5}&-10&0&0&1250&25\\& & \color{blue}{-50} & & & \\ \hline &\color{blue}{-10}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -50 \right) } = \color{orangered}{ -50 } $
$$ \begin{array}{c|rrrrr}5&-10&\color{orangered}{ 0 }&0&1250&25\\& & \color{orangered}{-50} & & & \\ \hline &-10&\color{orangered}{-50}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ \left( -50 \right) } = \color{blue}{ -250 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{5}&-10&0&0&1250&25\\& & -50& \color{blue}{-250} & & \\ \hline &-10&\color{blue}{-50}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -250 \right) } = \color{orangered}{ -250 } $
$$ \begin{array}{c|rrrrr}5&-10&0&\color{orangered}{ 0 }&1250&25\\& & -50& \color{orangered}{-250} & & \\ \hline &-10&-50&\color{orangered}{-250}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ \left( -250 \right) } = \color{blue}{ -1250 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{5}&-10&0&0&1250&25\\& & -50& -250& \color{blue}{-1250} & \\ \hline &-10&-50&\color{blue}{-250}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 1250 } + \color{orangered}{ \left( -1250 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrr}5&-10&0&0&\color{orangered}{ 1250 }&25\\& & -50& -250& \color{orangered}{-1250} & \\ \hline &-10&-50&-250&\color{orangered}{0}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{5}&-10&0&0&1250&25\\& & -50& -250& -1250& \color{blue}{0} \\ \hline &-10&-50&-250&\color{blue}{0}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 25 } + \color{orangered}{ 0 } = \color{orangered}{ 25 } $
$$ \begin{array}{c|rrrrr}5&-10&0&0&1250&\color{orangered}{ 25 }\\& & -50& -250& -1250& \color{orangered}{0} \\ \hline &\color{blue}{-10}&\color{blue}{-50}&\color{blue}{-250}&\color{blue}{0}&\color{orangered}{25} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -10x^{3}-50x^{2}-250x } $ with a remainder of $ \color{red}{ 25 } $.