Divide using synthetic division $$ ( 4x^{3}-5x^{2}+4x-20 ) \div ( x-20 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}20&4&-5&4&-20\\& & 80& 1500& \color{black}{30080} \\ \hline &\color{blue}{4}&\color{blue}{75}&\color{blue}{1504}&\color{orangered}{30060} \end{array} $$The solution is:
$$ \frac{ 4x^{3}-5x^{2}+4x-20 }{ x-20 } = \color{blue}{4x^{2}+75x+1504} ~+~ \frac{ \color{red}{ 30060 } }{ x-20 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -20 = 0 $ ( $ x = \color{blue}{ 20 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{20}&4&-5&4&-20\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}20&\color{orangered}{ 4 }&-5&4&-20\\& & & & \\ \hline &\color{orangered}{4}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 20 } \cdot \color{blue}{ 4 } = \color{blue}{ 80 } $.
$$ \begin{array}{c|rrrr}\color{blue}{20}&4&-5&4&-20\\& & \color{blue}{80} & & \\ \hline &\color{blue}{4}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -5 } + \color{orangered}{ 80 } = \color{orangered}{ 75 } $
$$ \begin{array}{c|rrrr}20&4&\color{orangered}{ -5 }&4&-20\\& & \color{orangered}{80} & & \\ \hline &4&\color{orangered}{75}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 20 } \cdot \color{blue}{ 75 } = \color{blue}{ 1500 } $.
$$ \begin{array}{c|rrrr}\color{blue}{20}&4&-5&4&-20\\& & 80& \color{blue}{1500} & \\ \hline &4&\color{blue}{75}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 4 } + \color{orangered}{ 1500 } = \color{orangered}{ 1504 } $
$$ \begin{array}{c|rrrr}20&4&-5&\color{orangered}{ 4 }&-20\\& & 80& \color{orangered}{1500} & \\ \hline &4&75&\color{orangered}{1504}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 20 } \cdot \color{blue}{ 1504 } = \color{blue}{ 30080 } $.
$$ \begin{array}{c|rrrr}\color{blue}{20}&4&-5&4&-20\\& & 80& 1500& \color{blue}{30080} \\ \hline &4&75&\color{blue}{1504}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -20 } + \color{orangered}{ 30080 } = \color{orangered}{ 30060 } $
$$ \begin{array}{c|rrrr}20&4&-5&4&\color{orangered}{ -20 }\\& & 80& 1500& \color{orangered}{30080} \\ \hline &\color{blue}{4}&\color{blue}{75}&\color{blue}{1504}&\color{orangered}{30060} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 4x^{2}+75x+1504 } $ with a remainder of $ \color{red}{ 30060 } $.