Divide using synthetic division $$ ( 30x^{6}-32x^{5}-17x^{4}+17x^{3}+13x^{2}+8x+34 ) \div ( x-1 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrrr}1&30&-32&-17&17&13&8&34\\& & 30& -2& -19& -2& 11& \color{black}{19} \\ \hline &\color{blue}{30}&\color{blue}{-2}&\color{blue}{-19}&\color{blue}{-2}&\color{blue}{11}&\color{blue}{19}&\color{orangered}{53} \end{array} $$The solution is:
$$ \frac{ 30x^{6}-32x^{5}-17x^{4}+17x^{3}+13x^{2}+8x+34 }{ x-1 } = \color{blue}{30x^{5}-2x^{4}-19x^{3}-2x^{2}+11x+19} ~+~ \frac{ \color{red}{ 53 } }{ x-1 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -1 = 0 $ ( $ x = \color{blue}{ 1 } $ ) at the left.
$$ \begin{array}{c|rrrrrrr}\color{blue}{1}&30&-32&-17&17&13&8&34\\& & & & & & & \\ \hline &&&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrrr}1&\color{orangered}{ 30 }&-32&-17&17&13&8&34\\& & & & & & & \\ \hline &\color{orangered}{30}&&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 30 } = \color{blue}{ 30 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{1}&30&-32&-17&17&13&8&34\\& & \color{blue}{30} & & & & & \\ \hline &\color{blue}{30}&&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -32 } + \color{orangered}{ 30 } = \color{orangered}{ -2 } $
$$ \begin{array}{c|rrrrrrr}1&30&\color{orangered}{ -32 }&-17&17&13&8&34\\& & \color{orangered}{30} & & & & & \\ \hline &30&\color{orangered}{-2}&&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ \left( -2 \right) } = \color{blue}{ -2 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{1}&30&-32&-17&17&13&8&34\\& & 30& \color{blue}{-2} & & & & \\ \hline &30&\color{blue}{-2}&&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -17 } + \color{orangered}{ \left( -2 \right) } = \color{orangered}{ -19 } $
$$ \begin{array}{c|rrrrrrr}1&30&-32&\color{orangered}{ -17 }&17&13&8&34\\& & 30& \color{orangered}{-2} & & & & \\ \hline &30&-2&\color{orangered}{-19}&&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ \left( -19 \right) } = \color{blue}{ -19 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{1}&30&-32&-17&17&13&8&34\\& & 30& -2& \color{blue}{-19} & & & \\ \hline &30&-2&\color{blue}{-19}&&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 17 } + \color{orangered}{ \left( -19 \right) } = \color{orangered}{ -2 } $
$$ \begin{array}{c|rrrrrrr}1&30&-32&-17&\color{orangered}{ 17 }&13&8&34\\& & 30& -2& \color{orangered}{-19} & & & \\ \hline &30&-2&-19&\color{orangered}{-2}&&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ \left( -2 \right) } = \color{blue}{ -2 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{1}&30&-32&-17&17&13&8&34\\& & 30& -2& -19& \color{blue}{-2} & & \\ \hline &30&-2&-19&\color{blue}{-2}&&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 13 } + \color{orangered}{ \left( -2 \right) } = \color{orangered}{ 11 } $
$$ \begin{array}{c|rrrrrrr}1&30&-32&-17&17&\color{orangered}{ 13 }&8&34\\& & 30& -2& -19& \color{orangered}{-2} & & \\ \hline &30&-2&-19&-2&\color{orangered}{11}&& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 11 } = \color{blue}{ 11 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{1}&30&-32&-17&17&13&8&34\\& & 30& -2& -19& -2& \color{blue}{11} & \\ \hline &30&-2&-19&-2&\color{blue}{11}&& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 8 } + \color{orangered}{ 11 } = \color{orangered}{ 19 } $
$$ \begin{array}{c|rrrrrrr}1&30&-32&-17&17&13&\color{orangered}{ 8 }&34\\& & 30& -2& -19& -2& \color{orangered}{11} & \\ \hline &30&-2&-19&-2&11&\color{orangered}{19}& \end{array} $$Step 12 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 19 } = \color{blue}{ 19 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{1}&30&-32&-17&17&13&8&34\\& & 30& -2& -19& -2& 11& \color{blue}{19} \\ \hline &30&-2&-19&-2&11&\color{blue}{19}& \end{array} $$Step 13 : Add down last column: $ \color{orangered}{ 34 } + \color{orangered}{ 19 } = \color{orangered}{ 53 } $
$$ \begin{array}{c|rrrrrrr}1&30&-32&-17&17&13&8&\color{orangered}{ 34 }\\& & 30& -2& -19& -2& 11& \color{orangered}{19} \\ \hline &\color{blue}{30}&\color{blue}{-2}&\color{blue}{-19}&\color{blue}{-2}&\color{blue}{11}&\color{blue}{19}&\color{orangered}{53} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 30x^{5}-2x^{4}-19x^{3}-2x^{2}+11x+19 } $ with a remainder of $ \color{red}{ 53 } $.