Divide using synthetic division $$ ( x^{4}-10x^{3}+17x^{2}-7x+1 ) \div ( x-8 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}8&1&-10&17&-7&1\\& & 8& -16& 8& \color{black}{8} \\ \hline &\color{blue}{1}&\color{blue}{-2}&\color{blue}{1}&\color{blue}{1}&\color{orangered}{9} \end{array} $$The solution is:
$$ \frac{ x^{4}-10x^{3}+17x^{2}-7x+1 }{ x-8 } = \color{blue}{x^{3}-2x^{2}+x+1} ~+~ \frac{ \color{red}{ 9 } }{ x-8 } $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{8}&1&-10&17&-7&1\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}8&\color{orangered}{ 1 }&-10&17&-7&1\\& & & & & \\ \hline &\color{orangered}{1}&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{8}&1&-10&17&-7&1\\& & \color{blue}{8} & & & \\ \hline &\color{blue}{1}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -10 } + \color{orangered}{ 8 } = \color{orangered}{ -2 } $
$$ \begin{array}{c|rrrrr}8&1&\color{orangered}{ -10 }&17&-7&1\\& & \color{orangered}{8} & & & \\ \hline &1&\color{orangered}{-2}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 8 } \cdot \color{blue}{ \left( -2 \right) } = \color{blue}{ -16 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{8}&1&-10&17&-7&1\\& & 8& \color{blue}{-16} & & \\ \hline &1&\color{blue}{-2}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 17 } + \color{orangered}{ \left( -16 \right) } = \color{orangered}{ 1 } $
$$ \begin{array}{c|rrrrr}8&1&-10&\color{orangered}{ 17 }&-7&1\\& & 8& \color{orangered}{-16} & & \\ \hline &1&-2&\color{orangered}{1}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 8 } \cdot \color{blue}{ 1 } = \color{blue}{ 8 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{8}&1&-10&17&-7&1\\& & 8& -16& \color{blue}{8} & \\ \hline &1&-2&\color{blue}{1}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -7 } + \color{orangered}{ 8 } = \color{orangered}{ 1 } $
$$ \begin{array}{c|rrrrr}8&1&-10&17&\color{orangered}{ -7 }&1\\& & 8& -16& \color{orangered}{8} & \\ \hline &1&-2&1&\color{orangered}{1}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 8 } \cdot \color{blue}{ 1 } = \color{blue}{ 8 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{8}&1&-10&17&-7&1\\& & 8& -16& 8& \color{blue}{8} \\ \hline &1&-2&1&\color{blue}{1}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ 8 } = \color{orangered}{ 9 } $
$$ \begin{array}{c|rrrrr}8&1&-10&17&-7&\color{orangered}{ 1 }\\& & 8& -16& 8& \color{orangered}{8} \\ \hline &\color{blue}{1}&\color{blue}{-2}&\color{blue}{1}&\color{blue}{1}&\color{orangered}{9} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{3}-2x^{2}+x+1 } $ with a remainder of $ \color{red}{ 9 } $.