Divide using synthetic division $$ ( x^{4}-4096 ) \div ( x-8 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}8&1&0&0&0&-4096\\& & 8& 64& 512& \color{black}{4096} \\ \hline &\color{blue}{1}&\color{blue}{8}&\color{blue}{64}&\color{blue}{512}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ x^{4}-4096 }{ x-8 } = \color{blue}{x^{3}+8x^{2}+64x+512} $$Explanation
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.
$$ \begin{array}{c|rrrrr}\color{blue}{8}&1&0&0&0&-4096\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}8&\color{orangered}{ 1 }&0&0&0&-4096\\& & & & & \\ \hline &\color{orangered}{1}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 8 } \cdot \color{blue}{ 1 } = \color{blue}{ 8 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{8}&1&0&0&0&-4096\\& & \color{blue}{8} & & & \\ \hline &\color{blue}{1}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 8 } = \color{orangered}{ 8 } $
$$ \begin{array}{c|rrrrr}8&1&\color{orangered}{ 0 }&0&0&-4096\\& & \color{orangered}{8} & & & \\ \hline &1&\color{orangered}{8}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 8 } \cdot \color{blue}{ 8 } = \color{blue}{ 64 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{8}&1&0&0&0&-4096\\& & 8& \color{blue}{64} & & \\ \hline &1&\color{blue}{8}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 64 } = \color{orangered}{ 64 } $
$$ \begin{array}{c|rrrrr}8&1&0&\color{orangered}{ 0 }&0&-4096\\& & 8& \color{orangered}{64} & & \\ \hline &1&8&\color{orangered}{64}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 8 } \cdot \color{blue}{ 64 } = \color{blue}{ 512 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{8}&1&0&0&0&-4096\\& & 8& 64& \color{blue}{512} & \\ \hline &1&8&\color{blue}{64}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 512 } = \color{orangered}{ 512 } $
$$ \begin{array}{c|rrrrr}8&1&0&0&\color{orangered}{ 0 }&-4096\\& & 8& 64& \color{orangered}{512} & \\ \hline &1&8&64&\color{orangered}{512}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 8 } \cdot \color{blue}{ 512 } = \color{blue}{ 4096 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{8}&1&0&0&0&-4096\\& & 8& 64& 512& \color{blue}{4096} \\ \hline &1&8&64&\color{blue}{512}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -4096 } + \color{orangered}{ 4096 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrr}8&1&0&0&0&\color{orangered}{ -4096 }\\& & 8& 64& 512& \color{orangered}{4096} \\ \hline &\color{blue}{1}&\color{blue}{8}&\color{blue}{64}&\color{blue}{512}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{3}+8x^{2}+64x+512 } $ with a remainder of $ \color{red}{ 0 } $.