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