Divide using synthetic division $$ ( 6x^{5}-4x^{4}+2x^{2}+7 ) \div ( x-3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}3&6&-4&0&2&0&7\\& & 18& 42& 126& 384& \color{black}{1152} \\ \hline &\color{blue}{6}&\color{blue}{14}&\color{blue}{42}&\color{blue}{128}&\color{blue}{384}&\color{orangered}{1159} \end{array} $$The solution is:
$$ \frac{ 6x^{5}-4x^{4}+2x^{2}+7 }{ x-3 } = \color{blue}{6x^{4}+14x^{3}+42x^{2}+128x+384} ~+~ \frac{ \color{red}{ 1159 } }{ x-3 } $$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.
$$ \begin{array}{c|rrrrrr}\color{blue}{3}&6&-4&0&2&0&7\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}3&\color{orangered}{ 6 }&-4&0&2&0&7\\& & & & & & \\ \hline &\color{orangered}{6}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 6 } = \color{blue}{ 18 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{3}&6&-4&0&2&0&7\\& & \color{blue}{18} & & & & \\ \hline &\color{blue}{6}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -4 } + \color{orangered}{ 18 } = \color{orangered}{ 14 } $
$$ \begin{array}{c|rrrrrr}3&6&\color{orangered}{ -4 }&0&2&0&7\\& & \color{orangered}{18} & & & & \\ \hline &6&\color{orangered}{14}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 14 } = \color{blue}{ 42 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{3}&6&-4&0&2&0&7\\& & 18& \color{blue}{42} & & & \\ \hline &6&\color{blue}{14}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 42 } = \color{orangered}{ 42 } $
$$ \begin{array}{c|rrrrrr}3&6&-4&\color{orangered}{ 0 }&2&0&7\\& & 18& \color{orangered}{42} & & & \\ \hline &6&14&\color{orangered}{42}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 42 } = \color{blue}{ 126 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{3}&6&-4&0&2&0&7\\& & 18& 42& \color{blue}{126} & & \\ \hline &6&14&\color{blue}{42}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ 126 } = \color{orangered}{ 128 } $
$$ \begin{array}{c|rrrrrr}3&6&-4&0&\color{orangered}{ 2 }&0&7\\& & 18& 42& \color{orangered}{126} & & \\ \hline &6&14&42&\color{orangered}{128}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 128 } = \color{blue}{ 384 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{3}&6&-4&0&2&0&7\\& & 18& 42& 126& \color{blue}{384} & \\ \hline &6&14&42&\color{blue}{128}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 384 } = \color{orangered}{ 384 } $
$$ \begin{array}{c|rrrrrr}3&6&-4&0&2&\color{orangered}{ 0 }&7\\& & 18& 42& 126& \color{orangered}{384} & \\ \hline &6&14&42&128&\color{orangered}{384}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 384 } = \color{blue}{ 1152 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{3}&6&-4&0&2&0&7\\& & 18& 42& 126& 384& \color{blue}{1152} \\ \hline &6&14&42&128&\color{blue}{384}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 7 } + \color{orangered}{ 1152 } = \color{orangered}{ 1159 } $
$$ \begin{array}{c|rrrrrr}3&6&-4&0&2&0&\color{orangered}{ 7 }\\& & 18& 42& 126& 384& \color{orangered}{1152} \\ \hline &\color{blue}{6}&\color{blue}{14}&\color{blue}{42}&\color{blue}{128}&\color{blue}{384}&\color{orangered}{1159} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 6x^{4}+14x^{3}+42x^{2}+128x+384 } $ with a remainder of $ \color{red}{ 1159 } $.