Divide using synthetic division $$ ( 5x^{3}+8x^{2}-x+6 ) \div ( x-2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}2&5&8&-1&6\\& & 10& 36& \color{black}{70} \\ \hline &\color{blue}{5}&\color{blue}{18}&\color{blue}{35}&\color{orangered}{76} \end{array} $$The solution is:
$$ \frac{ 5x^{3}+8x^{2}-x+6 }{ x-2 } = \color{blue}{5x^{2}+18x+35} ~+~ \frac{ \color{red}{ 76 } }{ x-2 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -2 = 0 $ ( $ x = \color{blue}{ 2 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{2}&5&8&-1&6\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}2&\color{orangered}{ 5 }&8&-1&6\\& & & & \\ \hline &\color{orangered}{5}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 5 } = \color{blue}{ 10 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&5&8&-1&6\\& & \color{blue}{10} & & \\ \hline &\color{blue}{5}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 8 } + \color{orangered}{ 10 } = \color{orangered}{ 18 } $
$$ \begin{array}{c|rrrr}2&5&\color{orangered}{ 8 }&-1&6\\& & \color{orangered}{10} & & \\ \hline &5&\color{orangered}{18}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 18 } = \color{blue}{ 36 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&5&8&-1&6\\& & 10& \color{blue}{36} & \\ \hline &5&\color{blue}{18}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ 36 } = \color{orangered}{ 35 } $
$$ \begin{array}{c|rrrr}2&5&8&\color{orangered}{ -1 }&6\\& & 10& \color{orangered}{36} & \\ \hline &5&18&\color{orangered}{35}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 35 } = \color{blue}{ 70 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&5&8&-1&6\\& & 10& 36& \color{blue}{70} \\ \hline &5&18&\color{blue}{35}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 6 } + \color{orangered}{ 70 } = \color{orangered}{ 76 } $
$$ \begin{array}{c|rrrr}2&5&8&-1&\color{orangered}{ 6 }\\& & 10& 36& \color{orangered}{70} \\ \hline &\color{blue}{5}&\color{blue}{18}&\color{blue}{35}&\color{orangered}{76} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 5x^{2}+18x+35 } $ with a remainder of $ \color{red}{ 76 } $.