Divide using synthetic division $$ ( x^{3}+250x^{2}+100x ) \div ( 25x+9 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-9&1&250&100&0\\& & -9& -2169& \color{black}{18621} \\ \hline &\color{blue}{1}&\color{blue}{241}&\color{blue}{-2069}&\color{orangered}{18621} \end{array} $$The solution is:
$$ \frac{ x^{3}+250x^{2}+100x }{ x+9 } = \color{blue}{x^{2}+241x-2069} ~+~ \frac{ \color{red}{ 18621 } }{ x+9 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 9 = 0 $ ( $ x = \color{blue}{ -9 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{-9}&1&250&100&0\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-9&\color{orangered}{ 1 }&250&100&0\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -9 } \cdot \color{blue}{ 1 } = \color{blue}{ -9 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-9}&1&250&100&0\\& & \color{blue}{-9} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 250 } + \color{orangered}{ \left( -9 \right) } = \color{orangered}{ 241 } $
$$ \begin{array}{c|rrrr}-9&1&\color{orangered}{ 250 }&100&0\\& & \color{orangered}{-9} & & \\ \hline &1&\color{orangered}{241}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -9 } \cdot \color{blue}{ 241 } = \color{blue}{ -2169 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-9}&1&250&100&0\\& & -9& \color{blue}{-2169} & \\ \hline &1&\color{blue}{241}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 100 } + \color{orangered}{ \left( -2169 \right) } = \color{orangered}{ -2069 } $
$$ \begin{array}{c|rrrr}-9&1&250&\color{orangered}{ 100 }&0\\& & -9& \color{orangered}{-2169} & \\ \hline &1&241&\color{orangered}{-2069}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -9 } \cdot \color{blue}{ \left( -2069 \right) } = \color{blue}{ 18621 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-9}&1&250&100&0\\& & -9& -2169& \color{blue}{18621} \\ \hline &1&241&\color{blue}{-2069}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 18621 } = \color{orangered}{ 18621 } $
$$ \begin{array}{c|rrrr}-9&1&250&100&\color{orangered}{ 0 }\\& & -9& -2169& \color{orangered}{18621} \\ \hline &\color{blue}{1}&\color{blue}{241}&\color{blue}{-2069}&\color{orangered}{18621} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{2}+241x-2069 } $ with a remainder of $ \color{red}{ 18621 } $.