Divide using synthetic division $$ ( 6x^{3}+4x^{2}-4x+1 ) \div ( 3x-1 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}1&6&4&-4&1\\& & 6& 10& \color{black}{6} \\ \hline &\color{blue}{6}&\color{blue}{10}&\color{blue}{6}&\color{orangered}{7} \end{array} $$The solution is:
$$ \frac{ 6x^{3}+4x^{2}-4x+1 }{ x-1 } = \color{blue}{6x^{2}+10x+6} ~+~ \frac{ \color{red}{ 7 } }{ x-1 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -1 = 0 $ ( $ x = \color{blue}{ 1 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{1}&6&4&-4&1\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}1&\color{orangered}{ 6 }&4&-4&1\\& & & & \\ \hline &\color{orangered}{6}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 6 } = \color{blue}{ 6 } $.
$$ \begin{array}{c|rrrr}\color{blue}{1}&6&4&-4&1\\& & \color{blue}{6} & & \\ \hline &\color{blue}{6}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 4 } + \color{orangered}{ 6 } = \color{orangered}{ 10 } $
$$ \begin{array}{c|rrrr}1&6&\color{orangered}{ 4 }&-4&1\\& & \color{orangered}{6} & & \\ \hline &6&\color{orangered}{10}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 10 } = \color{blue}{ 10 } $.
$$ \begin{array}{c|rrrr}\color{blue}{1}&6&4&-4&1\\& & 6& \color{blue}{10} & \\ \hline &6&\color{blue}{10}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -4 } + \color{orangered}{ 10 } = \color{orangered}{ 6 } $
$$ \begin{array}{c|rrrr}1&6&4&\color{orangered}{ -4 }&1\\& & 6& \color{orangered}{10} & \\ \hline &6&10&\color{orangered}{6}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 6 } = \color{blue}{ 6 } $.
$$ \begin{array}{c|rrrr}\color{blue}{1}&6&4&-4&1\\& & 6& 10& \color{blue}{6} \\ \hline &6&10&\color{blue}{6}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ 6 } = \color{orangered}{ 7 } $
$$ \begin{array}{c|rrrr}1&6&4&-4&\color{orangered}{ 1 }\\& & 6& 10& \color{orangered}{6} \\ \hline &\color{blue}{6}&\color{blue}{10}&\color{blue}{6}&\color{orangered}{7} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 6x^{2}+10x+6 } $ with a remainder of $ \color{red}{ 7 } $.