Divide using synthetic division $$ ( x^{3}+17x^{2}+70x-54 ) \div ( x-18 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}18&1&17&70&-54\\& & 18& 630& \color{black}{12600} \\ \hline &\color{blue}{1}&\color{blue}{35}&\color{blue}{700}&\color{orangered}{12546} \end{array} $$The solution is:
$$ \frac{ x^{3}+17x^{2}+70x-54 }{ x-18 } = \color{blue}{x^{2}+35x+700} ~+~ \frac{ \color{red}{ 12546 } }{ x-18 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -18 = 0 $ ( $ x = \color{blue}{ 18 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{18}&1&17&70&-54\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}18&\color{orangered}{ 1 }&17&70&-54\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 18 } \cdot \color{blue}{ 1 } = \color{blue}{ 18 } $.
$$ \begin{array}{c|rrrr}\color{blue}{18}&1&17&70&-54\\& & \color{blue}{18} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 17 } + \color{orangered}{ 18 } = \color{orangered}{ 35 } $
$$ \begin{array}{c|rrrr}18&1&\color{orangered}{ 17 }&70&-54\\& & \color{orangered}{18} & & \\ \hline &1&\color{orangered}{35}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 18 } \cdot \color{blue}{ 35 } = \color{blue}{ 630 } $.
$$ \begin{array}{c|rrrr}\color{blue}{18}&1&17&70&-54\\& & 18& \color{blue}{630} & \\ \hline &1&\color{blue}{35}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 70 } + \color{orangered}{ 630 } = \color{orangered}{ 700 } $
$$ \begin{array}{c|rrrr}18&1&17&\color{orangered}{ 70 }&-54\\& & 18& \color{orangered}{630} & \\ \hline &1&35&\color{orangered}{700}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 18 } \cdot \color{blue}{ 700 } = \color{blue}{ 12600 } $.
$$ \begin{array}{c|rrrr}\color{blue}{18}&1&17&70&-54\\& & 18& 630& \color{blue}{12600} \\ \hline &1&35&\color{blue}{700}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -54 } + \color{orangered}{ 12600 } = \color{orangered}{ 12546 } $
$$ \begin{array}{c|rrrr}18&1&17&70&\color{orangered}{ -54 }\\& & 18& 630& \color{orangered}{12600} \\ \hline &\color{blue}{1}&\color{blue}{35}&\color{blue}{700}&\color{orangered}{12546} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{2}+35x+700 } $ with a remainder of $ \color{red}{ 12546 } $.