Divide using synthetic division $$ ( 3x^{3}-4x+31 ) \div ( x-6 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}6&3&0&-4&31\\& & 18& 108& \color{black}{624} \\ \hline &\color{blue}{3}&\color{blue}{18}&\color{blue}{104}&\color{orangered}{655} \end{array} $$The solution is:
$$ \frac{ 3x^{3}-4x+31 }{ x-6 } = \color{blue}{3x^{2}+18x+104} ~+~ \frac{ \color{red}{ 655 } }{ x-6 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -6 = 0 $ ( $ x = \color{blue}{ 6 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{6}&3&0&-4&31\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}6&\color{orangered}{ 3 }&0&-4&31\\& & & & \\ \hline &\color{orangered}{3}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ 3 } = \color{blue}{ 18 } $.
$$ \begin{array}{c|rrrr}\color{blue}{6}&3&0&-4&31\\& & \color{blue}{18} & & \\ \hline &\color{blue}{3}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 18 } = \color{orangered}{ 18 } $
$$ \begin{array}{c|rrrr}6&3&\color{orangered}{ 0 }&-4&31\\& & \color{orangered}{18} & & \\ \hline &3&\color{orangered}{18}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ 18 } = \color{blue}{ 108 } $.
$$ \begin{array}{c|rrrr}\color{blue}{6}&3&0&-4&31\\& & 18& \color{blue}{108} & \\ \hline &3&\color{blue}{18}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -4 } + \color{orangered}{ 108 } = \color{orangered}{ 104 } $
$$ \begin{array}{c|rrrr}6&3&0&\color{orangered}{ -4 }&31\\& & 18& \color{orangered}{108} & \\ \hline &3&18&\color{orangered}{104}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ 104 } = \color{blue}{ 624 } $.
$$ \begin{array}{c|rrrr}\color{blue}{6}&3&0&-4&31\\& & 18& 108& \color{blue}{624} \\ \hline &3&18&\color{blue}{104}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 31 } + \color{orangered}{ 624 } = \color{orangered}{ 655 } $
$$ \begin{array}{c|rrrr}6&3&0&-4&\color{orangered}{ 31 }\\& & 18& 108& \color{orangered}{624} \\ \hline &\color{blue}{3}&\color{blue}{18}&\color{blue}{104}&\color{orangered}{655} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{2}+18x+104 } $ with a remainder of $ \color{red}{ 655 } $.