Divide using synthetic division $$ ( x^{2}+2x+3 ) \div ( x-2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}2&1&2&3\\& & 2& \color{black}{8} \\ \hline &\color{blue}{1}&\color{blue}{4}&\color{orangered}{11} \end{array} $$The solution is:
$$ \frac{ x^{2}+2x+3 }{ x-2 } = \color{blue}{x+4} ~+~ \frac{ \color{red}{ 11 } }{ x-2 } $$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.
$$ \begin{array}{c|rrr}\color{blue}{2}&1&2&3\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}2&\color{orangered}{ 1 }&2&3\\& & & \\ \hline &\color{orangered}{1}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 1 } = \color{blue}{ 2 } $.
$$ \begin{array}{c|rrr}\color{blue}{2}&1&2&3\\& & \color{blue}{2} & \\ \hline &\color{blue}{1}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ 2 } = \color{orangered}{ 4 } $
$$ \begin{array}{c|rrr}2&1&\color{orangered}{ 2 }&3\\& & \color{orangered}{2} & \\ \hline &1&\color{orangered}{4}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 4 } = \color{blue}{ 8 } $.
$$ \begin{array}{c|rrr}\color{blue}{2}&1&2&3\\& & 2& \color{blue}{8} \\ \hline &1&\color{blue}{4}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ 8 } = \color{orangered}{ 11 } $
$$ \begin{array}{c|rrr}2&1&2&\color{orangered}{ 3 }\\& & 2& \color{orangered}{8} \\ \hline &\color{blue}{1}&\color{blue}{4}&\color{orangered}{11} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x+4 } $ with a remainder of $ \color{red}{ 11 } $.