Divide using synthetic division $$ ( x^{3}+17x^{2}+70x-54 ) \div ( x-3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}3&1&17&70&-54\\& & 3& 60& \color{black}{390} \\ \hline &\color{blue}{1}&\color{blue}{20}&\color{blue}{130}&\color{orangered}{336} \end{array} $$The solution is:
$$ \frac{ x^{3}+17x^{2}+70x-54 }{ x-3 } = \color{blue}{x^{2}+20x+130} ~+~ \frac{ \color{red}{ 336 } }{ x-3 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{3}&1&17&70&-54\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}3&\color{orangered}{ 1 }&17&70&-54\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 1 } = \color{blue}{ 3 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&1&17&70&-54\\& & \color{blue}{3} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 17 } + \color{orangered}{ 3 } = \color{orangered}{ 20 } $
$$ \begin{array}{c|rrrr}3&1&\color{orangered}{ 17 }&70&-54\\& & \color{orangered}{3} & & \\ \hline &1&\color{orangered}{20}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 20 } = \color{blue}{ 60 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&1&17&70&-54\\& & 3& \color{blue}{60} & \\ \hline &1&\color{blue}{20}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 70 } + \color{orangered}{ 60 } = \color{orangered}{ 130 } $
$$ \begin{array}{c|rrrr}3&1&17&\color{orangered}{ 70 }&-54\\& & 3& \color{orangered}{60} & \\ \hline &1&20&\color{orangered}{130}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 130 } = \color{blue}{ 390 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&1&17&70&-54\\& & 3& 60& \color{blue}{390} \\ \hline &1&20&\color{blue}{130}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -54 } + \color{orangered}{ 390 } = \color{orangered}{ 336 } $
$$ \begin{array}{c|rrrr}3&1&17&70&\color{orangered}{ -54 }\\& & 3& 60& \color{orangered}{390} \\ \hline &\color{blue}{1}&\color{blue}{20}&\color{blue}{130}&\color{orangered}{336} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{2}+20x+130 } $ with a remainder of $ \color{red}{ 336 } $.