Divide using synthetic division $$ ( 3x^{3}+13x^{2}+37x+2 ) \div ( x-3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}3&3&13&37&2\\& & 9& 66& \color{black}{309} \\ \hline &\color{blue}{3}&\color{blue}{22}&\color{blue}{103}&\color{orangered}{311} \end{array} $$The solution is:
$$ \frac{ 3x^{3}+13x^{2}+37x+2 }{ x-3 } = \color{blue}{3x^{2}+22x+103} ~+~ \frac{ \color{red}{ 311 } }{ x-3 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{3}&3&13&37&2\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}3&\color{orangered}{ 3 }&13&37&2\\& & & & \\ \hline &\color{orangered}{3}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 3 } = \color{blue}{ 9 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&3&13&37&2\\& & \color{blue}{9} & & \\ \hline &\color{blue}{3}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 13 } + \color{orangered}{ 9 } = \color{orangered}{ 22 } $
$$ \begin{array}{c|rrrr}3&3&\color{orangered}{ 13 }&37&2\\& & \color{orangered}{9} & & \\ \hline &3&\color{orangered}{22}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 22 } = \color{blue}{ 66 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&3&13&37&2\\& & 9& \color{blue}{66} & \\ \hline &3&\color{blue}{22}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 37 } + \color{orangered}{ 66 } = \color{orangered}{ 103 } $
$$ \begin{array}{c|rrrr}3&3&13&\color{orangered}{ 37 }&2\\& & 9& \color{orangered}{66} & \\ \hline &3&22&\color{orangered}{103}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 103 } = \color{blue}{ 309 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&3&13&37&2\\& & 9& 66& \color{blue}{309} \\ \hline &3&22&\color{blue}{103}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ 309 } = \color{orangered}{ 311 } $
$$ \begin{array}{c|rrrr}3&3&13&37&\color{orangered}{ 2 }\\& & 9& 66& \color{orangered}{309} \\ \hline &\color{blue}{3}&\color{blue}{22}&\color{blue}{103}&\color{orangered}{311} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{2}+22x+103 } $ with a remainder of $ \color{red}{ 311 } $.