Divide using synthetic division $$ ( 3x^{3}-10x^{2}+9x-2 ) \div ( x-0.333 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}\frac{ 333 }{ 1000 }&3&-10&9&-2\\& & \frac{ 999 }{ 1000 }& -\frac{ 2997333 }{ 1000000 }& \color{black}{\frac{ 1998888111 }{ 1000000000 }} \\ \hline &\color{blue}{3}&\color{blue}{-\frac{ 9001 }{ 1000 }}&\color{blue}{\frac{ 6002667 }{ 1000000 }}&\color{orangered}{-\frac{ 1111889 }{ 1000000000 }} \end{array} $$The solution is:
$$ \frac{ 3x^{3}-10x^{2}+9x-2 }{ x-\frac{ 333 }{ 1000 } } = \color{blue}{3x^{2}-\frac{ 9001 }{ 1000 }x+\frac{ 6002667 }{ 1000000 }} \color{red}{~-~} \frac{ \color{red}{ \frac{ 1111889 }{ 1000000000 } } }{ x-\frac{ 333 }{ 1000 } } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -\frac{ 333 }{ 1000 } = 0 $ ( $ x = \color{blue}{ \frac{ 333 }{ 1000 } } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{\frac{ 333 }{ 1000 }}&3&-10&9&-2\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}\frac{ 333 }{ 1000 }&\color{orangered}{ 3 }&-10&9&-2\\& & & & \\ \hline &\color{orangered}{3}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ \frac{ 333 }{ 1000 } } \cdot \color{blue}{ 3 } = \color{blue}{ \frac{ 999 }{ 1000 } } $.
$$ \begin{array}{c|rrrr}\color{blue}{\frac{ 333 }{ 1000 }}&3&-10&9&-2\\& & \color{blue}{\frac{ 999 }{ 1000 }} & & \\ \hline &\color{blue}{3}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -10 } + \color{orangered}{ \frac{ 999 }{ 1000 } } = \color{orangered}{ -\frac{ 9001 }{ 1000 } } $
$$ \begin{array}{c|rrrr}\frac{ 333 }{ 1000 }&3&\color{orangered}{ -10 }&9&-2\\& & \color{orangered}{\frac{ 999 }{ 1000 }} & & \\ \hline &3&\color{orangered}{-\frac{ 9001 }{ 1000 }}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ \frac{ 333 }{ 1000 } } \cdot \color{blue}{ \left( -\frac{ 9001 }{ 1000 } \right) } = \color{blue}{ -\frac{ 2997333 }{ 1000000 } } $.
$$ \begin{array}{c|rrrr}\color{blue}{\frac{ 333 }{ 1000 }}&3&-10&9&-2\\& & \frac{ 999 }{ 1000 }& \color{blue}{-\frac{ 2997333 }{ 1000000 }} & \\ \hline &3&\color{blue}{-\frac{ 9001 }{ 1000 }}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 9 } + \color{orangered}{ \left( -\frac{ 2997333 }{ 1000000 } \right) } = \color{orangered}{ \frac{ 6002667 }{ 1000000 } } $
$$ \begin{array}{c|rrrr}\frac{ 333 }{ 1000 }&3&-10&\color{orangered}{ 9 }&-2\\& & \frac{ 999 }{ 1000 }& \color{orangered}{-\frac{ 2997333 }{ 1000000 }} & \\ \hline &3&-\frac{ 9001 }{ 1000 }&\color{orangered}{\frac{ 6002667 }{ 1000000 }}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ \frac{ 333 }{ 1000 } } \cdot \color{blue}{ \frac{ 6002667 }{ 1000000 } } = \color{blue}{ \frac{ 1998888111 }{ 1000000000 } } $.
$$ \begin{array}{c|rrrr}\color{blue}{\frac{ 333 }{ 1000 }}&3&-10&9&-2\\& & \frac{ 999 }{ 1000 }& -\frac{ 2997333 }{ 1000000 }& \color{blue}{\frac{ 1998888111 }{ 1000000000 }} \\ \hline &3&-\frac{ 9001 }{ 1000 }&\color{blue}{\frac{ 6002667 }{ 1000000 }}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ \frac{ 1998888111 }{ 1000000000 } } = \color{orangered}{ -\frac{ 1111889 }{ 1000000000 } } $
$$ \begin{array}{c|rrrr}\frac{ 333 }{ 1000 }&3&-10&9&\color{orangered}{ -2 }\\& & \frac{ 999 }{ 1000 }& -\frac{ 2997333 }{ 1000000 }& \color{orangered}{\frac{ 1998888111 }{ 1000000000 }} \\ \hline &\color{blue}{3}&\color{blue}{-\frac{ 9001 }{ 1000 }}&\color{blue}{\frac{ 6002667 }{ 1000000 }}&\color{orangered}{-\frac{ 1111889 }{ 1000000000 }} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{2}-\frac{ 9001 }{ 1000 }x+\frac{ 6002667 }{ 1000000 } } $ with a remainder of $ \color{red}{ -\frac{ 1111889 }{ 1000000000 } } $.