Divide using synthetic division $$ ( 4x^{3}-3x^{2}-37x+10 ) \div ( x-5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}5&4&-3&-37&10\\& & 20& 85& \color{black}{240} \\ \hline &\color{blue}{4}&\color{blue}{17}&\color{blue}{48}&\color{orangered}{250} \end{array} $$The solution is:
$$ \frac{ 4x^{3}-3x^{2}-37x+10 }{ x-5 } = \color{blue}{4x^{2}+17x+48} ~+~ \frac{ \color{red}{ 250 } }{ x-5 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{5}&4&-3&-37&10\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}5&\color{orangered}{ 4 }&-3&-37&10\\& & & & \\ \hline &\color{orangered}{4}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 4 } = \color{blue}{ 20 } $.
$$ \begin{array}{c|rrrr}\color{blue}{5}&4&-3&-37&10\\& & \color{blue}{20} & & \\ \hline &\color{blue}{4}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ 20 } = \color{orangered}{ 17 } $
$$ \begin{array}{c|rrrr}5&4&\color{orangered}{ -3 }&-37&10\\& & \color{orangered}{20} & & \\ \hline &4&\color{orangered}{17}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 17 } = \color{blue}{ 85 } $.
$$ \begin{array}{c|rrrr}\color{blue}{5}&4&-3&-37&10\\& & 20& \color{blue}{85} & \\ \hline &4&\color{blue}{17}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -37 } + \color{orangered}{ 85 } = \color{orangered}{ 48 } $
$$ \begin{array}{c|rrrr}5&4&-3&\color{orangered}{ -37 }&10\\& & 20& \color{orangered}{85} & \\ \hline &4&17&\color{orangered}{48}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 48 } = \color{blue}{ 240 } $.
$$ \begin{array}{c|rrrr}\color{blue}{5}&4&-3&-37&10\\& & 20& 85& \color{blue}{240} \\ \hline &4&17&\color{blue}{48}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 10 } + \color{orangered}{ 240 } = \color{orangered}{ 250 } $
$$ \begin{array}{c|rrrr}5&4&-3&-37&\color{orangered}{ 10 }\\& & 20& 85& \color{orangered}{240} \\ \hline &\color{blue}{4}&\color{blue}{17}&\color{blue}{48}&\color{orangered}{250} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 4x^{2}+17x+48 } $ with a remainder of $ \color{red}{ 250 } $.