Divide using synthetic division $$ ( x^{5}+17x^{4}-28x^{3}+29x^{2}-19x ) \div ( x-1 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}1&1&17&-28&29&-19&0\\& & 1& 18& -10& 19& \color{black}{0} \\ \hline &\color{blue}{1}&\color{blue}{18}&\color{blue}{-10}&\color{blue}{19}&\color{blue}{0}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ x^{5}+17x^{4}-28x^{3}+29x^{2}-19x }{ x-1 } = \color{blue}{x^{4}+18x^{3}-10x^{2}+19x} $$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.
$$ \begin{array}{c|rrrrrr}\color{blue}{1}&1&17&-28&29&-19&0\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}1&\color{orangered}{ 1 }&17&-28&29&-19&0\\& & & & & & \\ \hline &\color{orangered}{1}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 1 } = \color{blue}{ 1 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{1}&1&17&-28&29&-19&0\\& & \color{blue}{1} & & & & \\ \hline &\color{blue}{1}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 17 } + \color{orangered}{ 1 } = \color{orangered}{ 18 } $
$$ \begin{array}{c|rrrrrr}1&1&\color{orangered}{ 17 }&-28&29&-19&0\\& & \color{orangered}{1} & & & & \\ \hline &1&\color{orangered}{18}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 18 } = \color{blue}{ 18 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{1}&1&17&-28&29&-19&0\\& & 1& \color{blue}{18} & & & \\ \hline &1&\color{blue}{18}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -28 } + \color{orangered}{ 18 } = \color{orangered}{ -10 } $
$$ \begin{array}{c|rrrrrr}1&1&17&\color{orangered}{ -28 }&29&-19&0\\& & 1& \color{orangered}{18} & & & \\ \hline &1&18&\color{orangered}{-10}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ \left( -10 \right) } = \color{blue}{ -10 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{1}&1&17&-28&29&-19&0\\& & 1& 18& \color{blue}{-10} & & \\ \hline &1&18&\color{blue}{-10}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 29 } + \color{orangered}{ \left( -10 \right) } = \color{orangered}{ 19 } $
$$ \begin{array}{c|rrrrrr}1&1&17&-28&\color{orangered}{ 29 }&-19&0\\& & 1& 18& \color{orangered}{-10} & & \\ \hline &1&18&-10&\color{orangered}{19}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 19 } = \color{blue}{ 19 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{1}&1&17&-28&29&-19&0\\& & 1& 18& -10& \color{blue}{19} & \\ \hline &1&18&-10&\color{blue}{19}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -19 } + \color{orangered}{ 19 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrrr}1&1&17&-28&29&\color{orangered}{ -19 }&0\\& & 1& 18& -10& \color{orangered}{19} & \\ \hline &1&18&-10&19&\color{orangered}{0}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{1}&1&17&-28&29&-19&0\\& & 1& 18& -10& 19& \color{blue}{0} \\ \hline &1&18&-10&19&\color{blue}{0}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 0 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrrr}1&1&17&-28&29&-19&\color{orangered}{ 0 }\\& & 1& 18& -10& 19& \color{orangered}{0} \\ \hline &\color{blue}{1}&\color{blue}{18}&\color{blue}{-10}&\color{blue}{19}&\color{blue}{0}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{4}+18x^{3}-10x^{2}+19x } $ with a remainder of $ \color{red}{ 0 } $.