Divide using synthetic division $$ ( 3x^{4}-2x^{3}-7x^{2}+4x+2 ) \div ( x-0.67 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}\frac{ 67 }{ 100 }&3&-2&-7&4&2\\& & \frac{ 201 }{ 100 }& \frac{ 67 }{ 10000 }& -\frac{ 4685511 }{ 1000000 }& \color{black}{-\frac{ 45929237 }{ 100000000 }} \\ \hline &\color{blue}{3}&\color{blue}{\frac{ 1 }{ 100 }}&\color{blue}{-\frac{ 69933 }{ 10000 }}&\color{blue}{-\frac{ 685511 }{ 1000000 }}&\color{orangered}{\frac{ 154070763 }{ 100000000 }} \end{array} $$The solution is:
$$ \frac{ 3x^{4}-2x^{3}-7x^{2}+4x+2 }{ x-\frac{ 67 }{ 100 } } = \color{blue}{3x^{3}+\frac{ 1 }{ 100 }x^{2}-\frac{ 69933 }{ 10000 }x-\frac{ 685511 }{ 1000000 }} ~+~ \frac{ \color{red}{ \frac{ 154070763 }{ 100000000 } } }{ x-\frac{ 67 }{ 100 } } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -\frac{ 67 }{ 100 } = 0 $ ( $ x = \color{blue}{ \frac{ 67 }{ 100 } } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{\frac{ 67 }{ 100 }}&3&-2&-7&4&2\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}\frac{ 67 }{ 100 }&\color{orangered}{ 3 }&-2&-7&4&2\\& & & & & \\ \hline &\color{orangered}{3}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ \frac{ 67 }{ 100 } } \cdot \color{blue}{ 3 } = \color{blue}{ \frac{ 201 }{ 100 } } $.
$$ \begin{array}{c|rrrrr}\color{blue}{\frac{ 67 }{ 100 }}&3&-2&-7&4&2\\& & \color{blue}{\frac{ 201 }{ 100 }} & & & \\ \hline &\color{blue}{3}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ \frac{ 201 }{ 100 } } = \color{orangered}{ \frac{ 1 }{ 100 } } $
$$ \begin{array}{c|rrrrr}\frac{ 67 }{ 100 }&3&\color{orangered}{ -2 }&-7&4&2\\& & \color{orangered}{\frac{ 201 }{ 100 }} & & & \\ \hline &3&\color{orangered}{\frac{ 1 }{ 100 }}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ \frac{ 67 }{ 100 } } \cdot \color{blue}{ \frac{ 1 }{ 100 } } = \color{blue}{ \frac{ 67 }{ 10000 } } $.
$$ \begin{array}{c|rrrrr}\color{blue}{\frac{ 67 }{ 100 }}&3&-2&-7&4&2\\& & \frac{ 201 }{ 100 }& \color{blue}{\frac{ 67 }{ 10000 }} & & \\ \hline &3&\color{blue}{\frac{ 1 }{ 100 }}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -7 } + \color{orangered}{ \frac{ 67 }{ 10000 } } = \color{orangered}{ -\frac{ 69933 }{ 10000 } } $
$$ \begin{array}{c|rrrrr}\frac{ 67 }{ 100 }&3&-2&\color{orangered}{ -7 }&4&2\\& & \frac{ 201 }{ 100 }& \color{orangered}{\frac{ 67 }{ 10000 }} & & \\ \hline &3&\frac{ 1 }{ 100 }&\color{orangered}{-\frac{ 69933 }{ 10000 }}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ \frac{ 67 }{ 100 } } \cdot \color{blue}{ \left( -\frac{ 69933 }{ 10000 } \right) } = \color{blue}{ -\frac{ 4685511 }{ 1000000 } } $.
$$ \begin{array}{c|rrrrr}\color{blue}{\frac{ 67 }{ 100 }}&3&-2&-7&4&2\\& & \frac{ 201 }{ 100 }& \frac{ 67 }{ 10000 }& \color{blue}{-\frac{ 4685511 }{ 1000000 }} & \\ \hline &3&\frac{ 1 }{ 100 }&\color{blue}{-\frac{ 69933 }{ 10000 }}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 4 } + \color{orangered}{ \left( -\frac{ 4685511 }{ 1000000 } \right) } = \color{orangered}{ -\frac{ 685511 }{ 1000000 } } $
$$ \begin{array}{c|rrrrr}\frac{ 67 }{ 100 }&3&-2&-7&\color{orangered}{ 4 }&2\\& & \frac{ 201 }{ 100 }& \frac{ 67 }{ 10000 }& \color{orangered}{-\frac{ 4685511 }{ 1000000 }} & \\ \hline &3&\frac{ 1 }{ 100 }&-\frac{ 69933 }{ 10000 }&\color{orangered}{-\frac{ 685511 }{ 1000000 }}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ \frac{ 67 }{ 100 } } \cdot \color{blue}{ \left( -\frac{ 685511 }{ 1000000 } \right) } = \color{blue}{ -\frac{ 45929237 }{ 100000000 } } $.
$$ \begin{array}{c|rrrrr}\color{blue}{\frac{ 67 }{ 100 }}&3&-2&-7&4&2\\& & \frac{ 201 }{ 100 }& \frac{ 67 }{ 10000 }& -\frac{ 4685511 }{ 1000000 }& \color{blue}{-\frac{ 45929237 }{ 100000000 }} \\ \hline &3&\frac{ 1 }{ 100 }&-\frac{ 69933 }{ 10000 }&\color{blue}{-\frac{ 685511 }{ 1000000 }}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ \left( -\frac{ 45929237 }{ 100000000 } \right) } = \color{orangered}{ \frac{ 154070763 }{ 100000000 } } $
$$ \begin{array}{c|rrrrr}\frac{ 67 }{ 100 }&3&-2&-7&4&\color{orangered}{ 2 }\\& & \frac{ 201 }{ 100 }& \frac{ 67 }{ 10000 }& -\frac{ 4685511 }{ 1000000 }& \color{orangered}{-\frac{ 45929237 }{ 100000000 }} \\ \hline &\color{blue}{3}&\color{blue}{\frac{ 1 }{ 100 }}&\color{blue}{-\frac{ 69933 }{ 10000 }}&\color{blue}{-\frac{ 685511 }{ 1000000 }}&\color{orangered}{\frac{ 154070763 }{ 100000000 }} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{3}+\frac{ 1 }{ 100 }x^{2}-\frac{ 69933 }{ 10000 }x-\frac{ 685511 }{ 1000000 } } $ with a remainder of $ \color{red}{ \frac{ 154070763 }{ 100000000 } } $.