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