Divide using synthetic division $$ ( 3x^{3}+5x^{2}-6x+80 ) \div ( x+1 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-1&3&5&-6&80\\& & -3& -2& \color{black}{8} \\ \hline &\color{blue}{3}&\color{blue}{2}&\color{blue}{-8}&\color{orangered}{88} \end{array} $$The solution is:
$$ \frac{ 3x^{3}+5x^{2}-6x+80 }{ x+1 } = \color{blue}{3x^{2}+2x-8} ~+~ \frac{ \color{red}{ 88 } }{ x+1 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&3&5&-6&80\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-1&\color{orangered}{ 3 }&5&-6&80\\& & & & \\ \hline &\color{orangered}{3}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 3 } = \color{blue}{ -3 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&3&5&-6&80\\& & \color{blue}{-3} & & \\ \hline &\color{blue}{3}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ \left( -3 \right) } = \color{orangered}{ 2 } $
$$ \begin{array}{c|rrrr}-1&3&\color{orangered}{ 5 }&-6&80\\& & \color{orangered}{-3} & & \\ \hline &3&\color{orangered}{2}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 2 } = \color{blue}{ -2 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&3&5&-6&80\\& & -3& \color{blue}{-2} & \\ \hline &3&\color{blue}{2}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -6 } + \color{orangered}{ \left( -2 \right) } = \color{orangered}{ -8 } $
$$ \begin{array}{c|rrrr}-1&3&5&\color{orangered}{ -6 }&80\\& & -3& \color{orangered}{-2} & \\ \hline &3&2&\color{orangered}{-8}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -8 \right) } = \color{blue}{ 8 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&3&5&-6&80\\& & -3& -2& \color{blue}{8} \\ \hline &3&2&\color{blue}{-8}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 80 } + \color{orangered}{ 8 } = \color{orangered}{ 88 } $
$$ \begin{array}{c|rrrr}-1&3&5&-6&\color{orangered}{ 80 }\\& & -3& -2& \color{orangered}{8} \\ \hline &\color{blue}{3}&\color{blue}{2}&\color{blue}{-8}&\color{orangered}{88} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{2}+2x-8 } $ with a remainder of $ \color{red}{ 88 } $.