Divide using synthetic division $$ ( x^{5}-5x^{4}+10x^{3}-10x^{2}+5x-1 ) \div ( x+3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}-3&1&-5&10&-10&5&-1\\& & -3& 24& -102& 336& \color{black}{-1023} \\ \hline &\color{blue}{1}&\color{blue}{-8}&\color{blue}{34}&\color{blue}{-112}&\color{blue}{341}&\color{orangered}{-1024} \end{array} $$The solution is:
$$ \frac{ x^{5}-5x^{4}+10x^{3}-10x^{2}+5x-1 }{ x+3 } = \color{blue}{x^{4}-8x^{3}+34x^{2}-112x+341} \color{red}{~-~} \frac{ \color{red}{ 1024 } }{ x+3 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 3 = 0 $ ( $ x = \color{blue}{ -3 } $ ) at the left.
$$ \begin{array}{c|rrrrrr}\color{blue}{-3}&1&-5&10&-10&5&-1\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}-3&\color{orangered}{ 1 }&-5&10&-10&5&-1\\& & & & & & \\ \hline &\color{orangered}{1}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 1 } = \color{blue}{ -3 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-3}&1&-5&10&-10&5&-1\\& & \color{blue}{-3} & & & & \\ \hline &\color{blue}{1}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -5 } + \color{orangered}{ \left( -3 \right) } = \color{orangered}{ -8 } $
$$ \begin{array}{c|rrrrrr}-3&1&\color{orangered}{ -5 }&10&-10&5&-1\\& & \color{orangered}{-3} & & & & \\ \hline &1&\color{orangered}{-8}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -8 \right) } = \color{blue}{ 24 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-3}&1&-5&10&-10&5&-1\\& & -3& \color{blue}{24} & & & \\ \hline &1&\color{blue}{-8}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 10 } + \color{orangered}{ 24 } = \color{orangered}{ 34 } $
$$ \begin{array}{c|rrrrrr}-3&1&-5&\color{orangered}{ 10 }&-10&5&-1\\& & -3& \color{orangered}{24} & & & \\ \hline &1&-8&\color{orangered}{34}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 34 } = \color{blue}{ -102 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-3}&1&-5&10&-10&5&-1\\& & -3& 24& \color{blue}{-102} & & \\ \hline &1&-8&\color{blue}{34}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -10 } + \color{orangered}{ \left( -102 \right) } = \color{orangered}{ -112 } $
$$ \begin{array}{c|rrrrrr}-3&1&-5&10&\color{orangered}{ -10 }&5&-1\\& & -3& 24& \color{orangered}{-102} & & \\ \hline &1&-8&34&\color{orangered}{-112}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -112 \right) } = \color{blue}{ 336 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-3}&1&-5&10&-10&5&-1\\& & -3& 24& -102& \color{blue}{336} & \\ \hline &1&-8&34&\color{blue}{-112}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ 336 } = \color{orangered}{ 341 } $
$$ \begin{array}{c|rrrrrr}-3&1&-5&10&-10&\color{orangered}{ 5 }&-1\\& & -3& 24& -102& \color{orangered}{336} & \\ \hline &1&-8&34&-112&\color{orangered}{341}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 341 } = \color{blue}{ -1023 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-3}&1&-5&10&-10&5&-1\\& & -3& 24& -102& 336& \color{blue}{-1023} \\ \hline &1&-8&34&-112&\color{blue}{341}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ \left( -1023 \right) } = \color{orangered}{ -1024 } $
$$ \begin{array}{c|rrrrrr}-3&1&-5&10&-10&5&\color{orangered}{ -1 }\\& & -3& 24& -102& 336& \color{orangered}{-1023} \\ \hline &\color{blue}{1}&\color{blue}{-8}&\color{blue}{34}&\color{blue}{-112}&\color{blue}{341}&\color{orangered}{-1024} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{4}-8x^{3}+34x^{2}-112x+341 } $ with a remainder of $ \color{red}{ -1024 } $.