Find remainder when $ -47x^{5}+868x^{4}-7946x^{3}+37021x^{2}-78407x+48510 $ is divided by $ x-9 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}9&-47&868&-7946&37021&-78407&48510\\& & -423& 4005& -35469& 13968& \color{black}{-579951} \\ \hline &\color{blue}{-47}&\color{blue}{445}&\color{blue}{-3941}&\color{blue}{1552}&\color{blue}{-64439}&\color{orangered}{-531441} \end{array} $$The remainder when $ -47x^{5}+868x^{4}-7946x^{3}+37021x^{2}-78407x+48510 $ is divided by $ x-9 $ is $ \, \color{red}{ -531441 } $.
Explanation
We can find remainder using synthetic division method.
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.
$$ \begin{array}{c|rrrrrr}\color{blue}{9}&-47&868&-7946&37021&-78407&48510\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}9&\color{orangered}{ -47 }&868&-7946&37021&-78407&48510\\& & & & & & \\ \hline &\color{orangered}{-47}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 9 } \cdot \color{blue}{ \left( -47 \right) } = \color{blue}{ -423 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{9}&-47&868&-7946&37021&-78407&48510\\& & \color{blue}{-423} & & & & \\ \hline &\color{blue}{-47}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 868 } + \color{orangered}{ \left( -423 \right) } = \color{orangered}{ 445 } $
$$ \begin{array}{c|rrrrrr}9&-47&\color{orangered}{ 868 }&-7946&37021&-78407&48510\\& & \color{orangered}{-423} & & & & \\ \hline &-47&\color{orangered}{445}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 9 } \cdot \color{blue}{ 445 } = \color{blue}{ 4005 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{9}&-47&868&-7946&37021&-78407&48510\\& & -423& \color{blue}{4005} & & & \\ \hline &-47&\color{blue}{445}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -7946 } + \color{orangered}{ 4005 } = \color{orangered}{ -3941 } $
$$ \begin{array}{c|rrrrrr}9&-47&868&\color{orangered}{ -7946 }&37021&-78407&48510\\& & -423& \color{orangered}{4005} & & & \\ \hline &-47&445&\color{orangered}{-3941}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 9 } \cdot \color{blue}{ \left( -3941 \right) } = \color{blue}{ -35469 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{9}&-47&868&-7946&37021&-78407&48510\\& & -423& 4005& \color{blue}{-35469} & & \\ \hline &-47&445&\color{blue}{-3941}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 37021 } + \color{orangered}{ \left( -35469 \right) } = \color{orangered}{ 1552 } $
$$ \begin{array}{c|rrrrrr}9&-47&868&-7946&\color{orangered}{ 37021 }&-78407&48510\\& & -423& 4005& \color{orangered}{-35469} & & \\ \hline &-47&445&-3941&\color{orangered}{1552}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 9 } \cdot \color{blue}{ 1552 } = \color{blue}{ 13968 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{9}&-47&868&-7946&37021&-78407&48510\\& & -423& 4005& -35469& \color{blue}{13968} & \\ \hline &-47&445&-3941&\color{blue}{1552}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -78407 } + \color{orangered}{ 13968 } = \color{orangered}{ -64439 } $
$$ \begin{array}{c|rrrrrr}9&-47&868&-7946&37021&\color{orangered}{ -78407 }&48510\\& & -423& 4005& -35469& \color{orangered}{13968} & \\ \hline &-47&445&-3941&1552&\color{orangered}{-64439}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 9 } \cdot \color{blue}{ \left( -64439 \right) } = \color{blue}{ -579951 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{9}&-47&868&-7946&37021&-78407&48510\\& & -423& 4005& -35469& 13968& \color{blue}{-579951} \\ \hline &-47&445&-3941&1552&\color{blue}{-64439}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 48510 } + \color{orangered}{ \left( -579951 \right) } = \color{orangered}{ -531441 } $
$$ \begin{array}{c|rrrrrr}9&-47&868&-7946&37021&-78407&\color{orangered}{ 48510 }\\& & -423& 4005& -35469& 13968& \color{orangered}{-579951} \\ \hline &\color{blue}{-47}&\color{blue}{445}&\color{blue}{-3941}&\color{blue}{1552}&\color{blue}{-64439}&\color{orangered}{-531441} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ -531441 }\right) $.