Divide using synthetic division $$ ( -2x^{4}-22x^{3}-37x^{2}+143x+70 ) \div ( x-2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}2&-2&-22&-37&143&70\\& & -4& -52& -178& \color{black}{-70} \\ \hline &\color{blue}{-2}&\color{blue}{-26}&\color{blue}{-89}&\color{blue}{-35}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ -2x^{4}-22x^{3}-37x^{2}+143x+70 }{ x-2 } = \color{blue}{-2x^{3}-26x^{2}-89x-35} $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{2}&-2&-22&-37&143&70\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}2&\color{orangered}{ -2 }&-22&-37&143&70\\& & & & & \\ \hline &\color{orangered}{-2}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -2 \right) } = \color{blue}{ -4 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{2}&-2&-22&-37&143&70\\& & \color{blue}{-4} & & & \\ \hline &\color{blue}{-2}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -22 } + \color{orangered}{ \left( -4 \right) } = \color{orangered}{ -26 } $
$$ \begin{array}{c|rrrrr}2&-2&\color{orangered}{ -22 }&-37&143&70\\& & \color{orangered}{-4} & & & \\ \hline &-2&\color{orangered}{-26}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -26 \right) } = \color{blue}{ -52 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{2}&-2&-22&-37&143&70\\& & -4& \color{blue}{-52} & & \\ \hline &-2&\color{blue}{-26}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -37 } + \color{orangered}{ \left( -52 \right) } = \color{orangered}{ -89 } $
$$ \begin{array}{c|rrrrr}2&-2&-22&\color{orangered}{ -37 }&143&70\\& & -4& \color{orangered}{-52} & & \\ \hline &-2&-26&\color{orangered}{-89}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -89 \right) } = \color{blue}{ -178 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{2}&-2&-22&-37&143&70\\& & -4& -52& \color{blue}{-178} & \\ \hline &-2&-26&\color{blue}{-89}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 143 } + \color{orangered}{ \left( -178 \right) } = \color{orangered}{ -35 } $
$$ \begin{array}{c|rrrrr}2&-2&-22&-37&\color{orangered}{ 143 }&70\\& & -4& -52& \color{orangered}{-178} & \\ \hline &-2&-26&-89&\color{orangered}{-35}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -35 \right) } = \color{blue}{ -70 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{2}&-2&-22&-37&143&70\\& & -4& -52& -178& \color{blue}{-70} \\ \hline &-2&-26&-89&\color{blue}{-35}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 70 } + \color{orangered}{ \left( -70 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrr}2&-2&-22&-37&143&\color{orangered}{ 70 }\\& & -4& -52& -178& \color{orangered}{-70} \\ \hline &\color{blue}{-2}&\color{blue}{-26}&\color{blue}{-89}&\color{blue}{-35}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -2x^{3}-26x^{2}-89x-35 } $ with a remainder of $ \color{red}{ 0 } $.