Divide using synthetic division $$ ( x^{3}-4x^{2}+17x-34 ) \div ( x-3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}3&1&-4&17&-34\\& & 3& -3& \color{black}{42} \\ \hline &\color{blue}{1}&\color{blue}{-1}&\color{blue}{14}&\color{orangered}{8} \end{array} $$The solution is:
$$ \frac{ x^{3}-4x^{2}+17x-34 }{ x-3 } = \color{blue}{x^{2}-x+14} ~+~ \frac{ \color{red}{ 8 } }{ x-3 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -3 = 0 $ ( $ x = \color{blue}{ 3 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{3}&1&-4&17&-34\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}3&\color{orangered}{ 1 }&-4&17&-34\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 1 } = \color{blue}{ 3 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&1&-4&17&-34\\& & \color{blue}{3} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -4 } + \color{orangered}{ 3 } = \color{orangered}{ -1 } $
$$ \begin{array}{c|rrrr}3&1&\color{orangered}{ -4 }&17&-34\\& & \color{orangered}{3} & & \\ \hline &1&\color{orangered}{-1}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ -3 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&1&-4&17&-34\\& & 3& \color{blue}{-3} & \\ \hline &1&\color{blue}{-1}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 17 } + \color{orangered}{ \left( -3 \right) } = \color{orangered}{ 14 } $
$$ \begin{array}{c|rrrr}3&1&-4&\color{orangered}{ 17 }&-34\\& & 3& \color{orangered}{-3} & \\ \hline &1&-1&\color{orangered}{14}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 14 } = \color{blue}{ 42 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&1&-4&17&-34\\& & 3& -3& \color{blue}{42} \\ \hline &1&-1&\color{blue}{14}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -34 } + \color{orangered}{ 42 } = \color{orangered}{ 8 } $
$$ \begin{array}{c|rrrr}3&1&-4&17&\color{orangered}{ -34 }\\& & 3& -3& \color{orangered}{42} \\ \hline &\color{blue}{1}&\color{blue}{-1}&\color{blue}{14}&\color{orangered}{8} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{2}-x+14 } $ with a remainder of $ \color{red}{ 8 } $.