Divide using synthetic division $$ ( 3x^{3}+2x^{2}-3x+8 ) \div ( x-2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}2&3&2&-3&8\\& & 6& 16& \color{black}{26} \\ \hline &\color{blue}{3}&\color{blue}{8}&\color{blue}{13}&\color{orangered}{34} \end{array} $$The solution is:
$$ \frac{ 3x^{3}+2x^{2}-3x+8 }{ x-2 } = \color{blue}{3x^{2}+8x+13} ~+~ \frac{ \color{red}{ 34 } }{ x-2 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{2}&3&2&-3&8\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}2&\color{orangered}{ 3 }&2&-3&8\\& & & & \\ \hline &\color{orangered}{3}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 3 } = \color{blue}{ 6 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&3&2&-3&8\\& & \color{blue}{6} & & \\ \hline &\color{blue}{3}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ 6 } = \color{orangered}{ 8 } $
$$ \begin{array}{c|rrrr}2&3&\color{orangered}{ 2 }&-3&8\\& & \color{orangered}{6} & & \\ \hline &3&\color{orangered}{8}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 8 } = \color{blue}{ 16 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&3&2&-3&8\\& & 6& \color{blue}{16} & \\ \hline &3&\color{blue}{8}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ 16 } = \color{orangered}{ 13 } $
$$ \begin{array}{c|rrrr}2&3&2&\color{orangered}{ -3 }&8\\& & 6& \color{orangered}{16} & \\ \hline &3&8&\color{orangered}{13}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 13 } = \color{blue}{ 26 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&3&2&-3&8\\& & 6& 16& \color{blue}{26} \\ \hline &3&8&\color{blue}{13}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 8 } + \color{orangered}{ 26 } = \color{orangered}{ 34 } $
$$ \begin{array}{c|rrrr}2&3&2&-3&\color{orangered}{ 8 }\\& & 6& 16& \color{orangered}{26} \\ \hline &\color{blue}{3}&\color{blue}{8}&\color{blue}{13}&\color{orangered}{34} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{2}+8x+13 } $ with a remainder of $ \color{red}{ 34 } $.