Divide using synthetic division $$ ( x^{3}+17x^{2}+70x-54 ) \div ( x-1 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}1&1&17&70&-54\\& & 1& 18& \color{black}{88} \\ \hline &\color{blue}{1}&\color{blue}{18}&\color{blue}{88}&\color{orangered}{34} \end{array} $$The solution is:
$$ \frac{ x^{3}+17x^{2}+70x-54 }{ x-1 } = \color{blue}{x^{2}+18x+88} ~+~ \frac{ \color{red}{ 34 } }{ x-1 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{1}&1&17&70&-54\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}1&\color{orangered}{ 1 }&17&70&-54\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrr}\color{blue}{1}&1&17&70&-54\\& & \color{blue}{1} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 17 } + \color{orangered}{ 1 } = \color{orangered}{ 18 } $
$$ \begin{array}{c|rrrr}1&1&\color{orangered}{ 17 }&70&-54\\& & \color{orangered}{1} & & \\ \hline &1&\color{orangered}{18}&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrr}\color{blue}{1}&1&17&70&-54\\& & 1& \color{blue}{18} & \\ \hline &1&\color{blue}{18}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 70 } + \color{orangered}{ 18 } = \color{orangered}{ 88 } $
$$ \begin{array}{c|rrrr}1&1&17&\color{orangered}{ 70 }&-54\\& & 1& \color{orangered}{18} & \\ \hline &1&18&\color{orangered}{88}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 88 } = \color{blue}{ 88 } $.
$$ \begin{array}{c|rrrr}\color{blue}{1}&1&17&70&-54\\& & 1& 18& \color{blue}{88} \\ \hline &1&18&\color{blue}{88}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -54 } + \color{orangered}{ 88 } = \color{orangered}{ 34 } $
$$ \begin{array}{c|rrrr}1&1&17&70&\color{orangered}{ -54 }\\& & 1& 18& \color{orangered}{88} \\ \hline &\color{blue}{1}&\color{blue}{18}&\color{blue}{88}&\color{orangered}{34} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{2}+18x+88 } $ with a remainder of $ \color{red}{ 34 } $.