Divide using synthetic division $$ ( 8x^{4}-4x^{3}+56x-19 ) \div ( 8x-4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}4&8&-4&0&56&-19\\& & 32& 112& 448& \color{black}{2016} \\ \hline &\color{blue}{8}&\color{blue}{28}&\color{blue}{112}&\color{blue}{504}&\color{orangered}{1997} \end{array} $$The solution is:
$$ \frac{ 8x^{4}-4x^{3}+56x-19 }{ x-4 } = \color{blue}{8x^{3}+28x^{2}+112x+504} ~+~ \frac{ \color{red}{ 1997 } }{ x-4 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -4 = 0 $ ( $ x = \color{blue}{ 4 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&8&-4&0&56&-19\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}4&\color{orangered}{ 8 }&-4&0&56&-19\\& & & & & \\ \hline &\color{orangered}{8}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 8 } = \color{blue}{ 32 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&8&-4&0&56&-19\\& & \color{blue}{32} & & & \\ \hline &\color{blue}{8}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -4 } + \color{orangered}{ 32 } = \color{orangered}{ 28 } $
$$ \begin{array}{c|rrrrr}4&8&\color{orangered}{ -4 }&0&56&-19\\& & \color{orangered}{32} & & & \\ \hline &8&\color{orangered}{28}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 28 } = \color{blue}{ 112 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&8&-4&0&56&-19\\& & 32& \color{blue}{112} & & \\ \hline &8&\color{blue}{28}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 112 } = \color{orangered}{ 112 } $
$$ \begin{array}{c|rrrrr}4&8&-4&\color{orangered}{ 0 }&56&-19\\& & 32& \color{orangered}{112} & & \\ \hline &8&28&\color{orangered}{112}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 112 } = \color{blue}{ 448 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&8&-4&0&56&-19\\& & 32& 112& \color{blue}{448} & \\ \hline &8&28&\color{blue}{112}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 56 } + \color{orangered}{ 448 } = \color{orangered}{ 504 } $
$$ \begin{array}{c|rrrrr}4&8&-4&0&\color{orangered}{ 56 }&-19\\& & 32& 112& \color{orangered}{448} & \\ \hline &8&28&112&\color{orangered}{504}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 504 } = \color{blue}{ 2016 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&8&-4&0&56&-19\\& & 32& 112& 448& \color{blue}{2016} \\ \hline &8&28&112&\color{blue}{504}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -19 } + \color{orangered}{ 2016 } = \color{orangered}{ 1997 } $
$$ \begin{array}{c|rrrrr}4&8&-4&0&56&\color{orangered}{ -19 }\\& & 32& 112& 448& \color{orangered}{2016} \\ \hline &\color{blue}{8}&\color{blue}{28}&\color{blue}{112}&\color{blue}{504}&\color{orangered}{1997} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 8x^{3}+28x^{2}+112x+504 } $ with a remainder of $ \color{red}{ 1997 } $.