Divide using synthetic division $$ ( 3x^{3}-x^{2}+2x+1 ) \div ( x-2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}2&3&-1&2&1\\& & 6& 10& \color{black}{24} \\ \hline &\color{blue}{3}&\color{blue}{5}&\color{blue}{12}&\color{orangered}{25} \end{array} $$The solution is:
$$ \frac{ 3x^{3}-x^{2}+2x+1 }{ x-2 } = \color{blue}{3x^{2}+5x+12} ~+~ \frac{ \color{red}{ 25 } }{ x-2 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{2}&3&-1&2&1\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}2&\color{orangered}{ 3 }&-1&2&1\\& & & & \\ \hline &\color{orangered}{3}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&3&-1&2&1\\& & \color{blue}{6} & & \\ \hline &\color{blue}{3}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ 6 } = \color{orangered}{ 5 } $
$$ \begin{array}{c|rrrr}2&3&\color{orangered}{ -1 }&2&1\\& & \color{orangered}{6} & & \\ \hline &3&\color{orangered}{5}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 5 } = \color{blue}{ 10 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&3&-1&2&1\\& & 6& \color{blue}{10} & \\ \hline &3&\color{blue}{5}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ 10 } = \color{orangered}{ 12 } $
$$ \begin{array}{c|rrrr}2&3&-1&\color{orangered}{ 2 }&1\\& & 6& \color{orangered}{10} & \\ \hline &3&5&\color{orangered}{12}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 12 } = \color{blue}{ 24 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&3&-1&2&1\\& & 6& 10& \color{blue}{24} \\ \hline &3&5&\color{blue}{12}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ 24 } = \color{orangered}{ 25 } $
$$ \begin{array}{c|rrrr}2&3&-1&2&\color{orangered}{ 1 }\\& & 6& 10& \color{orangered}{24} \\ \hline &\color{blue}{3}&\color{blue}{5}&\color{blue}{12}&\color{orangered}{25} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{2}+5x+12 } $ with a remainder of $ \color{red}{ 25 } $.