Divide using synthetic division $$ ( -3x^{4}+13x^{3}-17x^{2}-3x+10 ) \div ( x+0.67 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-\frac{ 67 }{ 100 }&-3&13&-17&-3&10\\& & \frac{ 201 }{ 100 }& -\frac{ 100567 }{ 10000 }& \frac{ 18127989 }{ 1000000 }& \color{black}{-\frac{ 1013575263 }{ 100000000 }} \\ \hline &\color{blue}{-3}&\color{blue}{\frac{ 1501 }{ 100 }}&\color{blue}{-\frac{ 270567 }{ 10000 }}&\color{blue}{\frac{ 15127989 }{ 1000000 }}&\color{orangered}{-\frac{ 13575263 }{ 100000000 }} \end{array} $$The solution is:
$$ \frac{ -3x^{4}+13x^{3}-17x^{2}-3x+10 }{ x+\frac{ 67 }{ 100 } } = \color{blue}{-3x^{3}+\frac{ 1501 }{ 100 }x^{2}-\frac{ 270567 }{ 10000 }x+\frac{ 15127989 }{ 1000000 }} \color{red}{~-~} \frac{ \color{red}{ \frac{ 13575263 }{ 100000000 } } }{ x+\frac{ 67 }{ 100 } } $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{-\frac{ 67 }{ 100 }}&-3&13&-17&-3&10\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-\frac{ 67 }{ 100 }&\color{orangered}{ -3 }&13&-17&-3&10\\& & & & & \\ \hline &\color{orangered}{-3}&&&& \end{array} $$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}{ \left( -3 \right) } = \color{blue}{ \frac{ 201 }{ 100 } } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-\frac{ 67 }{ 100 }}&-3&13&-17&-3&10\\& & \color{blue}{\frac{ 201 }{ 100 }} & & & \\ \hline &\color{blue}{-3}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 13 } + \color{orangered}{ \frac{ 201 }{ 100 } } = \color{orangered}{ \frac{ 1501 }{ 100 } } $
$$ \begin{array}{c|rrrrr}-\frac{ 67 }{ 100 }&-3&\color{orangered}{ 13 }&-17&-3&10\\& & \color{orangered}{\frac{ 201 }{ 100 }} & & & \\ \hline &-3&\color{orangered}{\frac{ 1501 }{ 100 }}&&& \end{array} $$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{ 1501 }{ 100 } } = \color{blue}{ -\frac{ 100567 }{ 10000 } } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-\frac{ 67 }{ 100 }}&-3&13&-17&-3&10\\& & \frac{ 201 }{ 100 }& \color{blue}{-\frac{ 100567 }{ 10000 }} & & \\ \hline &-3&\color{blue}{\frac{ 1501 }{ 100 }}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -17 } + \color{orangered}{ \left( -\frac{ 100567 }{ 10000 } \right) } = \color{orangered}{ -\frac{ 270567 }{ 10000 } } $
$$ \begin{array}{c|rrrrr}-\frac{ 67 }{ 100 }&-3&13&\color{orangered}{ -17 }&-3&10\\& & \frac{ 201 }{ 100 }& \color{orangered}{-\frac{ 100567 }{ 10000 }} & & \\ \hline &-3&\frac{ 1501 }{ 100 }&\color{orangered}{-\frac{ 270567 }{ 10000 }}&& \end{array} $$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{ 270567 }{ 10000 } \right) } = \color{blue}{ \frac{ 18127989 }{ 1000000 } } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-\frac{ 67 }{ 100 }}&-3&13&-17&-3&10\\& & \frac{ 201 }{ 100 }& -\frac{ 100567 }{ 10000 }& \color{blue}{\frac{ 18127989 }{ 1000000 }} & \\ \hline &-3&\frac{ 1501 }{ 100 }&\color{blue}{-\frac{ 270567 }{ 10000 }}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ \frac{ 18127989 }{ 1000000 } } = \color{orangered}{ \frac{ 15127989 }{ 1000000 } } $
$$ \begin{array}{c|rrrrr}-\frac{ 67 }{ 100 }&-3&13&-17&\color{orangered}{ -3 }&10\\& & \frac{ 201 }{ 100 }& -\frac{ 100567 }{ 10000 }& \color{orangered}{\frac{ 18127989 }{ 1000000 }} & \\ \hline &-3&\frac{ 1501 }{ 100 }&-\frac{ 270567 }{ 10000 }&\color{orangered}{\frac{ 15127989 }{ 1000000 }}& \end{array} $$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}{ \frac{ 15127989 }{ 1000000 } } = \color{blue}{ -\frac{ 1013575263 }{ 100000000 } } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-\frac{ 67 }{ 100 }}&-3&13&-17&-3&10\\& & \frac{ 201 }{ 100 }& -\frac{ 100567 }{ 10000 }& \frac{ 18127989 }{ 1000000 }& \color{blue}{-\frac{ 1013575263 }{ 100000000 }} \\ \hline &-3&\frac{ 1501 }{ 100 }&-\frac{ 270567 }{ 10000 }&\color{blue}{\frac{ 15127989 }{ 1000000 }}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 10 } + \color{orangered}{ \left( -\frac{ 1013575263 }{ 100000000 } \right) } = \color{orangered}{ -\frac{ 13575263 }{ 100000000 } } $
$$ \begin{array}{c|rrrrr}-\frac{ 67 }{ 100 }&-3&13&-17&-3&\color{orangered}{ 10 }\\& & \frac{ 201 }{ 100 }& -\frac{ 100567 }{ 10000 }& \frac{ 18127989 }{ 1000000 }& \color{orangered}{-\frac{ 1013575263 }{ 100000000 }} \\ \hline &\color{blue}{-3}&\color{blue}{\frac{ 1501 }{ 100 }}&\color{blue}{-\frac{ 270567 }{ 10000 }}&\color{blue}{\frac{ 15127989 }{ 1000000 }}&\color{orangered}{-\frac{ 13575263 }{ 100000000 }} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -3x^{3}+\frac{ 1501 }{ 100 }x^{2}-\frac{ 270567 }{ 10000 }x+\frac{ 15127989 }{ 1000000 } } $ with a remainder of $ \color{red}{ -\frac{ 13575263 }{ 100000000 } } $.