Divide using synthetic division $$ ( -5x^{7}+3x^{3}-2x^{2} ) \div ( x+1 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrrrrrrr}-1&-5&0&0&0&3&-2&0&0\\& & 5& -5& 5& -5& 2& 0& \color{black}{0} \\ \hline &\color{blue}{-5}&\color{blue}{5}&\color{blue}{-5}&\color{blue}{5}&\color{blue}{-2}&\color{blue}{0}&\color{blue}{0}&\color{orangered}{0} \end{array} $$The solution is:
$$ \dfrac{ -5x^{7}+3x^{3}-2x^{2} }{ x+1 } = \color{blue}{-5x^{6}+5x^{5}-5x^{4}+5x^{3}-2x^{2}} $$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.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{-1}&-5&0&0&0&3&-2&0&0\\& & & & & & & & \\ \hline &&&&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrrrr}-1&\color{orangered}{ -5 }&0&0&0&3&-2&0&0\\& & & & & & & & \\ \hline &\color{orangered}{-5}&&&&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{-1}&-5&0&0&0&3&-2&0&0\\& & \color{blue}{5} & & & & & & \\ \hline &\color{blue}{-5}&&&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 5 } = \color{orangered}{ 5 } $
$$ \begin{array}{c|rrrrrrrr}-1&-5&\color{orangered}{ 0 }&0&0&3&-2&0&0\\& & \color{orangered}{5} & & & & & & \\ \hline &-5&\color{orangered}{5}&&&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 5 } = \color{blue}{ -5 } $.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{-1}&-5&0&0&0&3&-2&0&0\\& & 5& \color{blue}{-5} & & & & & \\ \hline &-5&\color{blue}{5}&&&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -5 \right) } = \color{orangered}{ -5 } $
$$ \begin{array}{c|rrrrrrrr}-1&-5&0&\color{orangered}{ 0 }&0&3&-2&0&0\\& & 5& \color{orangered}{-5} & & & & & \\ \hline &-5&5&\color{orangered}{-5}&&&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -5 \right) } = \color{blue}{ 5 } $.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{-1}&-5&0&0&0&3&-2&0&0\\& & 5& -5& \color{blue}{5} & & & & \\ \hline &-5&5&\color{blue}{-5}&&&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 5 } = \color{orangered}{ 5 } $
$$ \begin{array}{c|rrrrrrrr}-1&-5&0&0&\color{orangered}{ 0 }&3&-2&0&0\\& & 5& -5& \color{orangered}{5} & & & & \\ \hline &-5&5&-5&\color{orangered}{5}&&&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 5 } = \color{blue}{ -5 } $.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{-1}&-5&0&0&0&3&-2&0&0\\& & 5& -5& 5& \color{blue}{-5} & & & \\ \hline &-5&5&-5&\color{blue}{5}&&&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ \left( -5 \right) } = \color{orangered}{ -2 } $
$$ \begin{array}{c|rrrrrrrr}-1&-5&0&0&0&\color{orangered}{ 3 }&-2&0&0\\& & 5& -5& 5& \color{orangered}{-5} & & & \\ \hline &-5&5&-5&5&\color{orangered}{-2}&&& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -2 \right) } = \color{blue}{ 2 } $.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{-1}&-5&0&0&0&3&-2&0&0\\& & 5& -5& 5& -5& \color{blue}{2} & & \\ \hline &-5&5&-5&5&\color{blue}{-2}&&& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ 2 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrrrrr}-1&-5&0&0&0&3&\color{orangered}{ -2 }&0&0\\& & 5& -5& 5& -5& \color{orangered}{2} & & \\ \hline &-5&5&-5&5&-2&\color{orangered}{0}&& \end{array} $$Step 12 : 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 } $.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{-1}&-5&0&0&0&3&-2&0&0\\& & 5& -5& 5& -5& 2& \color{blue}{0} & \\ \hline &-5&5&-5&5&-2&\color{blue}{0}&& \end{array} $$Step 13 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 0 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrrrrr}-1&-5&0&0&0&3&-2&\color{orangered}{ 0 }&0\\& & 5& -5& 5& -5& 2& \color{orangered}{0} & \\ \hline &-5&5&-5&5&-2&0&\color{orangered}{0}& \end{array} $$Step 14 : 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 } $.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{-1}&-5&0&0&0&3&-2&0&0\\& & 5& -5& 5& -5& 2& 0& \color{blue}{0} \\ \hline &-5&5&-5&5&-2&0&\color{blue}{0}& \end{array} $$Step 15 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 0 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrrrrr}-1&-5&0&0&0&3&-2&0&\color{orangered}{ 0 }\\& & 5& -5& 5& -5& 2& 0& \color{orangered}{0} \\ \hline &\color{blue}{-5}&\color{blue}{5}&\color{blue}{-5}&\color{blue}{5}&\color{blue}{-2}&\color{blue}{0}&\color{blue}{0}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -5x^{6}+5x^{5}-5x^{4}+5x^{3}-2x^{2} } $ with a remainder of $ \color{red}{ 0 } $.