Divide using synthetic division $$ ( 2x^{3}+3x^{2}-8x+3 ) \div ( x-3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}3&2&3&-8&3\\& & 6& 27& \color{black}{57} \\ \hline &\color{blue}{2}&\color{blue}{9}&\color{blue}{19}&\color{orangered}{60} \end{array} $$The solution is:
$$ \frac{ 2x^{3}+3x^{2}-8x+3 }{ x-3 } = \color{blue}{2x^{2}+9x+19} ~+~ \frac{ \color{red}{ 60 } }{ x-3 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{3}&2&3&-8&3\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}3&\color{orangered}{ 2 }&3&-8&3\\& & & & \\ \hline &\color{orangered}{2}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 2 } = \color{blue}{ 6 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&2&3&-8&3\\& & \color{blue}{6} & & \\ \hline &\color{blue}{2}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ 6 } = \color{orangered}{ 9 } $
$$ \begin{array}{c|rrrr}3&2&\color{orangered}{ 3 }&-8&3\\& & \color{orangered}{6} & & \\ \hline &2&\color{orangered}{9}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 9 } = \color{blue}{ 27 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&2&3&-8&3\\& & 6& \color{blue}{27} & \\ \hline &2&\color{blue}{9}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -8 } + \color{orangered}{ 27 } = \color{orangered}{ 19 } $
$$ \begin{array}{c|rrrr}3&2&3&\color{orangered}{ -8 }&3\\& & 6& \color{orangered}{27} & \\ \hline &2&9&\color{orangered}{19}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 19 } = \color{blue}{ 57 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&2&3&-8&3\\& & 6& 27& \color{blue}{57} \\ \hline &2&9&\color{blue}{19}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ 57 } = \color{orangered}{ 60 } $
$$ \begin{array}{c|rrrr}3&2&3&-8&\color{orangered}{ 3 }\\& & 6& 27& \color{orangered}{57} \\ \hline &\color{blue}{2}&\color{blue}{9}&\color{blue}{19}&\color{orangered}{60} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x^{2}+9x+19 } $ with a remainder of $ \color{red}{ 60 } $.