Find remainder when $ 4x^{2}-6x-4 $ is divided by $ 2x-8 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}8&4&-6&-4\\& & 32& \color{black}{208} \\ \hline &\color{blue}{4}&\color{blue}{26}&\color{orangered}{204} \end{array} $$The remainder when $ 4x^{2}-6x-4 $ is divided by $ x-8 $ is $ \, \color{red}{ 204 } $.
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 -8 = 0 $ ( $ x = \color{blue}{ 8 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{8}&4&-6&-4\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}8&\color{orangered}{ 4 }&-6&-4\\& & & \\ \hline &\color{orangered}{4}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 8 } \cdot \color{blue}{ 4 } = \color{blue}{ 32 } $.
$$ \begin{array}{c|rrr}\color{blue}{8}&4&-6&-4\\& & \color{blue}{32} & \\ \hline &\color{blue}{4}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -6 } + \color{orangered}{ 32 } = \color{orangered}{ 26 } $
$$ \begin{array}{c|rrr}8&4&\color{orangered}{ -6 }&-4\\& & \color{orangered}{32} & \\ \hline &4&\color{orangered}{26}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 8 } \cdot \color{blue}{ 26 } = \color{blue}{ 208 } $.
$$ \begin{array}{c|rrr}\color{blue}{8}&4&-6&-4\\& & 32& \color{blue}{208} \\ \hline &4&\color{blue}{26}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -4 } + \color{orangered}{ 208 } = \color{orangered}{ 204 } $
$$ \begin{array}{c|rrr}8&4&-6&\color{orangered}{ -4 }\\& & 32& \color{orangered}{208} \\ \hline &\color{blue}{4}&\color{blue}{26}&\color{orangered}{204} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 204 }\right) $.