Divide using synthetic division $$ ( -10x^{6}-x^{5}+7x^{4}+5x^{3}-3x^{2}-3x+2 ) \div ( x+2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrrr}-2&-10&-1&7&5&-3&-3&2\\& & 20& -38& 62& -134& 274& \color{black}{-542} \\ \hline &\color{blue}{-10}&\color{blue}{19}&\color{blue}{-31}&\color{blue}{67}&\color{blue}{-137}&\color{blue}{271}&\color{orangered}{-540} \end{array} $$The solution is:
$$ \frac{ -10x^{6}-x^{5}+7x^{4}+5x^{3}-3x^{2}-3x+2 }{ x+2 } = \color{blue}{-10x^{5}+19x^{4}-31x^{3}+67x^{2}-137x+271} \color{red}{~-~} \frac{ \color{red}{ 540 } }{ x+2 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 2 = 0 $ ( $ x = \color{blue}{ -2 } $ ) at the left.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-2}&-10&-1&7&5&-3&-3&2\\& & & & & & & \\ \hline &&&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrrr}-2&\color{orangered}{ -10 }&-1&7&5&-3&-3&2\\& & & & & & & \\ \hline &\color{orangered}{-10}&&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -10 \right) } = \color{blue}{ 20 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-2}&-10&-1&7&5&-3&-3&2\\& & \color{blue}{20} & & & & & \\ \hline &\color{blue}{-10}&&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ 20 } = \color{orangered}{ 19 } $
$$ \begin{array}{c|rrrrrrr}-2&-10&\color{orangered}{ -1 }&7&5&-3&-3&2\\& & \color{orangered}{20} & & & & & \\ \hline &-10&\color{orangered}{19}&&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 19 } = \color{blue}{ -38 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-2}&-10&-1&7&5&-3&-3&2\\& & 20& \color{blue}{-38} & & & & \\ \hline &-10&\color{blue}{19}&&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 7 } + \color{orangered}{ \left( -38 \right) } = \color{orangered}{ -31 } $
$$ \begin{array}{c|rrrrrrr}-2&-10&-1&\color{orangered}{ 7 }&5&-3&-3&2\\& & 20& \color{orangered}{-38} & & & & \\ \hline &-10&19&\color{orangered}{-31}&&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -31 \right) } = \color{blue}{ 62 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-2}&-10&-1&7&5&-3&-3&2\\& & 20& -38& \color{blue}{62} & & & \\ \hline &-10&19&\color{blue}{-31}&&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ 62 } = \color{orangered}{ 67 } $
$$ \begin{array}{c|rrrrrrr}-2&-10&-1&7&\color{orangered}{ 5 }&-3&-3&2\\& & 20& -38& \color{orangered}{62} & & & \\ \hline &-10&19&-31&\color{orangered}{67}&&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 67 } = \color{blue}{ -134 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-2}&-10&-1&7&5&-3&-3&2\\& & 20& -38& 62& \color{blue}{-134} & & \\ \hline &-10&19&-31&\color{blue}{67}&&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ \left( -134 \right) } = \color{orangered}{ -137 } $
$$ \begin{array}{c|rrrrrrr}-2&-10&-1&7&5&\color{orangered}{ -3 }&-3&2\\& & 20& -38& 62& \color{orangered}{-134} & & \\ \hline &-10&19&-31&67&\color{orangered}{-137}&& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -137 \right) } = \color{blue}{ 274 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-2}&-10&-1&7&5&-3&-3&2\\& & 20& -38& 62& -134& \color{blue}{274} & \\ \hline &-10&19&-31&67&\color{blue}{-137}&& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ 274 } = \color{orangered}{ 271 } $
$$ \begin{array}{c|rrrrrrr}-2&-10&-1&7&5&-3&\color{orangered}{ -3 }&2\\& & 20& -38& 62& -134& \color{orangered}{274} & \\ \hline &-10&19&-31&67&-137&\color{orangered}{271}& \end{array} $$Step 12 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 271 } = \color{blue}{ -542 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-2}&-10&-1&7&5&-3&-3&2\\& & 20& -38& 62& -134& 274& \color{blue}{-542} \\ \hline &-10&19&-31&67&-137&\color{blue}{271}& \end{array} $$Step 13 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ \left( -542 \right) } = \color{orangered}{ -540 } $
$$ \begin{array}{c|rrrrrrr}-2&-10&-1&7&5&-3&-3&\color{orangered}{ 2 }\\& & 20& -38& 62& -134& 274& \color{orangered}{-542} \\ \hline &\color{blue}{-10}&\color{blue}{19}&\color{blue}{-31}&\color{blue}{67}&\color{blue}{-137}&\color{blue}{271}&\color{orangered}{-540} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -10x^{5}+19x^{4}-31x^{3}+67x^{2}-137x+271 } $ with a remainder of $ \color{red}{ -540 } $.