Divide using synthetic division $$ ( -47x^{5}+868x^{4}-7946x^{3}+37021x^{2}-78407x+48510 ) \div ( x-5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}5&-47&868&-7946&37021&-78407&48510\\& & -235& 3165& -23905& 65580& \color{black}{-64135} \\ \hline &\color{blue}{-47}&\color{blue}{633}&\color{blue}{-4781}&\color{blue}{13116}&\color{blue}{-12827}&\color{orangered}{-15625} \end{array} $$The solution is:
$$ \frac{ -47x^{5}+868x^{4}-7946x^{3}+37021x^{2}-78407x+48510 }{ x-5 } = \color{blue}{-47x^{4}+633x^{3}-4781x^{2}+13116x-12827} \color{red}{~-~} \frac{ \color{red}{ 15625 } }{ x-5 } $$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.
$$ \begin{array}{c|rrrrrr}\color{blue}{5}&-47&868&-7946&37021&-78407&48510\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}5&\color{orangered}{ -47 }&868&-7946&37021&-78407&48510\\& & & & & & \\ \hline &\color{orangered}{-47}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ \left( -47 \right) } = \color{blue}{ -235 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{5}&-47&868&-7946&37021&-78407&48510\\& & \color{blue}{-235} & & & & \\ \hline &\color{blue}{-47}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 868 } + \color{orangered}{ \left( -235 \right) } = \color{orangered}{ 633 } $
$$ \begin{array}{c|rrrrrr}5&-47&\color{orangered}{ 868 }&-7946&37021&-78407&48510\\& & \color{orangered}{-235} & & & & \\ \hline &-47&\color{orangered}{633}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 633 } = \color{blue}{ 3165 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{5}&-47&868&-7946&37021&-78407&48510\\& & -235& \color{blue}{3165} & & & \\ \hline &-47&\color{blue}{633}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -7946 } + \color{orangered}{ 3165 } = \color{orangered}{ -4781 } $
$$ \begin{array}{c|rrrrrr}5&-47&868&\color{orangered}{ -7946 }&37021&-78407&48510\\& & -235& \color{orangered}{3165} & & & \\ \hline &-47&633&\color{orangered}{-4781}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ \left( -4781 \right) } = \color{blue}{ -23905 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{5}&-47&868&-7946&37021&-78407&48510\\& & -235& 3165& \color{blue}{-23905} & & \\ \hline &-47&633&\color{blue}{-4781}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 37021 } + \color{orangered}{ \left( -23905 \right) } = \color{orangered}{ 13116 } $
$$ \begin{array}{c|rrrrrr}5&-47&868&-7946&\color{orangered}{ 37021 }&-78407&48510\\& & -235& 3165& \color{orangered}{-23905} & & \\ \hline &-47&633&-4781&\color{orangered}{13116}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 13116 } = \color{blue}{ 65580 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{5}&-47&868&-7946&37021&-78407&48510\\& & -235& 3165& -23905& \color{blue}{65580} & \\ \hline &-47&633&-4781&\color{blue}{13116}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -78407 } + \color{orangered}{ 65580 } = \color{orangered}{ -12827 } $
$$ \begin{array}{c|rrrrrr}5&-47&868&-7946&37021&\color{orangered}{ -78407 }&48510\\& & -235& 3165& -23905& \color{orangered}{65580} & \\ \hline &-47&633&-4781&13116&\color{orangered}{-12827}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ \left( -12827 \right) } = \color{blue}{ -64135 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{5}&-47&868&-7946&37021&-78407&48510\\& & -235& 3165& -23905& 65580& \color{blue}{-64135} \\ \hline &-47&633&-4781&13116&\color{blue}{-12827}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 48510 } + \color{orangered}{ \left( -64135 \right) } = \color{orangered}{ -15625 } $
$$ \begin{array}{c|rrrrrr}5&-47&868&-7946&37021&-78407&\color{orangered}{ 48510 }\\& & -235& 3165& -23905& 65580& \color{orangered}{-64135} \\ \hline &\color{blue}{-47}&\color{blue}{633}&\color{blue}{-4781}&\color{blue}{13116}&\color{blue}{-12827}&\color{orangered}{-15625} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -47x^{4}+633x^{3}-4781x^{2}+13116x-12827 } $ with a remainder of $ \color{red}{ -15625 } $.