Divide using synthetic division $$ ( 3x^{3}-13x^{2}+7x-7 ) \div ( x+1 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-1&3&-13&7&-7\\& & -3& 16& \color{black}{-23} \\ \hline &\color{blue}{3}&\color{blue}{-16}&\color{blue}{23}&\color{orangered}{-30} \end{array} $$The solution is:
$$ \frac{ 3x^{3}-13x^{2}+7x-7 }{ x+1 } = \color{blue}{3x^{2}-16x+23} \color{red}{~-~} \frac{ \color{red}{ 30 } }{ x+1 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&3&-13&7&-7\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-1&\color{orangered}{ 3 }&-13&7&-7\\& & & & \\ \hline &\color{orangered}{3}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&3&-13&7&-7\\& & \color{blue}{-3} & & \\ \hline &\color{blue}{3}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -13 } + \color{orangered}{ \left( -3 \right) } = \color{orangered}{ -16 } $
$$ \begin{array}{c|rrrr}-1&3&\color{orangered}{ -13 }&7&-7\\& & \color{orangered}{-3} & & \\ \hline &3&\color{orangered}{-16}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -16 \right) } = \color{blue}{ 16 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&3&-13&7&-7\\& & -3& \color{blue}{16} & \\ \hline &3&\color{blue}{-16}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 7 } + \color{orangered}{ 16 } = \color{orangered}{ 23 } $
$$ \begin{array}{c|rrrr}-1&3&-13&\color{orangered}{ 7 }&-7\\& & -3& \color{orangered}{16} & \\ \hline &3&-16&\color{orangered}{23}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 23 } = \color{blue}{ -23 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&3&-13&7&-7\\& & -3& 16& \color{blue}{-23} \\ \hline &3&-16&\color{blue}{23}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -7 } + \color{orangered}{ \left( -23 \right) } = \color{orangered}{ -30 } $
$$ \begin{array}{c|rrrr}-1&3&-13&7&\color{orangered}{ -7 }\\& & -3& 16& \color{orangered}{-23} \\ \hline &\color{blue}{3}&\color{blue}{-16}&\color{blue}{23}&\color{orangered}{-30} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{2}-16x+23 } $ with a remainder of $ \color{red}{ -30 } $.